# On the diagonals of rational functions: the minimal number of variables

S Hassani, J Maillard, N Zenine

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S Hassani, J Maillard, N Zenine. On the diagonals of rational functions: the minimal number of variables. Journal of Physics A: Mathematical and Theoretical, 2025, 58 (25), pp.255202. 10.1088/1751-8121/adde02. hal-05146667

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# On the diagonals of rational functions: the minimal number of variables

To cite this article: S Hassani et al 2025 *J. Phys. A: Math. Theor.* **58** 255202

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# On the diagonals of rational functions: the minimal number of variables

S Hassani<fnref n="1" />, J M Maillard<fnref n="2" />,* and N Zenine<fnref n="3" />

E-mail: maillard@lptmc.jussieu.fr, jean-marie.maillard@sorbonne-universite.fr, okbi_h@yahoo.fr and njzenine@yahoo.com

Received 7 February 2025; revised 15 May 2025
Accepted for publication 28 May 2025
Published 18 June 2025

<figure><img src="image_2.png" /></figure>

CrossMark

## Abstract

From some observations on the linear differential operators occurring in the Lattice Green function of the *d*-dimensional face centered and simple cubic lattices, and on the linear differential operators occurring in the *n*-particle contributions $\chi^{(n)}$ to the magnetic susceptibility of the square Ising model, we forward some conjectures on the diagonals of rational functions. These conjectures are also in agreement with exact results we obtain for many Calabi–Yau operators, and many other examples related, or not related to physics. Consider a globally bounded power series which is the diagonal of rational functions of a certain number of variables, annihilated by an irreducible minimal order linear differential operator homomorphic to its adjoint. Among the logarithmic formal series solutions, at the origin, of this operator, call *n* the highest power of the logarithm. We conjecture that this diagonal series can be represented as a diagonal of a rational function with a minimal number of variables $N_v$ related to this highest power *n* by the relation $N_v = n + 2$. Since the operator is homomorphic to its adjoint, its differential Galois group is symplectic or orthogonal. We also conjecture that the symplectic or orthogonal character of the differential Galois group is related to the parity of the highest power *n*, namely symplectic for *n* odd and orthogonal for *n* even. We also sketch the case where the denominator of the rational function is not irreducible and is the product of, for instance, two polynomials. We recall that the linear differential operators occurring in the *n*-particle contributions $\chi^{(n)}$ to the magnetic susceptibility of the square Ising model factorize in a large number of direct sums

<figure><img src="image_3.png" /></figure>

<footnote n="1">¹ Retired: Centre de Recherche Nucléaire d’Alger, Alger, Algeria</footnote>

<footnote n="2">² LPTMC, UMR 7600 CNRS, Sorbonne Université, Tour 23, 5ème étage, case 121, 4 Place Jussieu, 75252 Paris Cedex 05, Paris, France</footnote>

<footnote n="3">³ Centre de Recherche Nucléaire d’Alger, 2 Bd. Frantz Fanon, BP 399, 16000 Alger, Algeria</footnote>

<footnote>* Author to whom any correspondence should be addressed.</footnote>
and products of factors. The analysis of the linear differential operators annihilating the diagonal of rational function where the denominator is the product of two polynomials, sheds some light on the emergence of such mixture of direct sums and products of factors. The conjecture $N_v = n+2$ still holds for such reducible linear differential operators.

Keywords: diagonal of rational function, multivariate series expansions, lattice Green function, Calabi–Yau operators, homomorphisms of linear differential operators, differential Galois groups, magnetic susceptibility of the square Ising model

# 1. Introduction

Diagonals of rational functions have been seen to occur naturally [1] for $n$-fold integrals in physics, field theory, enumerative combinatorics [2], etc. On many examples and cases, striking properties emerged that are worthy to be understood.

When we seek for a characterization of the diagonal of rational functions representation of a D-finite *globally bounded* power series$^4$, one may think of the *least number of variables* $N_v$ occurring in the rational function$^5$.

Given a diagonal of a rational function $R_1$ of $N$ variables (which is necessarily [1] a D-finite *globally bounded* power series) there is a rational function $R$ with a *minimal number* $N_v$ of variables. One aim of this paper amounts to showing that this number of variables $N_v \le N$ is simply related to the logarithmic singular behavior, at the origin, of the formal series of the (minimal order) linear differential operator$^6$ annihilating the diagonal.

We illustrate our conjecture with the analysis of the lattice Green function (LGF) of the $d$-dimensional simple cubic (s.c) and face centered cubic (f.c.c) lattices [5–9], as well as results obtained for many Calabi–Yau operators [10], and an accumulation of other examples related, or not related, to physics, displayed (or not displayed) in this paper.

The differential operators for LGF of the s.c and f.c.c lattices are *irreducible*. These differential operators are *homomorphic to their adjoint* and, consequently, their differential Galois groups are (included in) symplectic or orthogonal groups. All these LGFs can, obviously, be cast into diagonal of rational functions.

This irreducibility is in sharp contrast with the differential operators of the $n$-particle contributions $\chi^{(n)7}$ to the magnetic susceptibility of the square Ising model [11], which have a large set of factors ([12] and references therein). Here, the symplectic, or orthogonal, character of the differential Galois group concerns the *factors* occurring in the factorization of the differential operators annihilating the $\chi^{(n)'}$'s. Furthermore, we observe, for the $n$-particle contributions $\chi^{(n)}$ of the susceptibility of the square Ising model, that, for each block of factors

<footnote>$^4$ A *globally bounded* [3] power series is a series with a finite radius of convergence, which can be recast into a series with *integer* coefficients by a unique rescaling of the variable $t \to \lambda \cdot t$. A diagonal of a rational function is necessarily [1] a D-finite *globally bounded* power series. Conversely, according to Christol’s conjecture [4], a D-finite *globally bounded* power series should be the diagonal of a rational function.</footnote>

<footnote>$^5$ It is obvious, that the rational function, whose diagonal gives a given *globally bounded* series, is far from unique. This series can be the diagonal of many rational functions, even with different numbers of variables.</footnote>

<footnote>$^6$ All the differential operators, in this paper, are linear. We will omit ‘linear’ in the sequel.</footnote>

<footnote>$^7$ The $\chi^{(n)}$ have a very convoluted form of algebraic fractions as integrands. They are shown to be diagonal of rational functions [1].</footnote>
in the differential operator which has a unique factorization, (e.g. for a block of three factors $L_n \cdot L_p \cdot L_q$), we have *alternately* orthogonal and symplectic groups.

We will show that these characteristics can be seen on the diagonals of rational functions, with simple enough expressions, that may lead to a better understanding of their occurrences.

With $P$ and $Q$ multivariate polynomials (with $Q(0, ..., 0) \neq 0$), the formal series of the (minimal order) differential operator annihilating the diagonal of the rational functions $P/Q^r$ (with $r$ an integer), correspond to a *finite dimensional* vectorial space related$^8$, as shown by Christol [13–15], to the de Rham cohomology. There is a *homomorphism* between the (minimal order) differential operators for the diagonal of $P/Q^k$ and for the diagonal of $1/Q$. Therefore, without too much loss of generality (see section 3 of [16]), we will basically restrict ourselves, in this paper, to rational functions in the form $R = 1/Q$, where $Q$ is an irreducible multivariate polynomial, with $Q(0, ..., 0) \neq 0$. We will also consider, for pedagogical reason, the case where the denominator $Q$ factors in only two polynomials, $Q = 1/Q_1/Q_2$, where at least one of the $Q_j$'s depends on all the variables$^9$.

In the sequel, we notice that, generically, for irreducible $Q$ over the rationals, the resulting (minimal order) differential operator, annihilating the diagonal of $1/Q$, seems to be systematically irreducible. In contrast, a factorization of the differential operator occurs for factorizable $Q$.

## 1.1. The formal solutions

The differential operators annihilating diagonals of rational functions are very selected differential operators [16]. The formal solutions, at the origin, of such differential operators (call it $L_q$) can be organized as the union of different sets (1) and (2), ... of formal series which makes the monodromy at $t = 0$ crystal clear$^{10}$:

$$ 
\begin{gathered}
S_0, \quad S_0 \cdot \ln(t) + S_{1,0}, \quad S_0 \cdot \frac{\ln(t)^2}{2!} + S_{1,0} \cdot \ln(t) + S_{2,0}, \\
\dots, \\
S_0 \cdot \frac{\ln(t)^n}{n!} + S_{1,0} \cdot \frac{\ln(t)^{n-1}}{(n-1)!} + S_{2,0} \cdot \frac{\ln(t)^{n-2}}{(n-2)!} + \dots + S_{n,0}, \quad (1)
\end{gathered}
$$

$$ 
\begin{gathered}
T_0, \quad T_0 \cdot \ln(t) + T_{1,0}, \quad T_0 \cdot \frac{\ln(t)^2}{2!} + T_{1,0} \cdot \ln(t) + T_{2,0}, \\
\dots, \\
T_0 \cdot \frac{\ln(t)^m}{m!} + T_{1,0} \cdot \frac{\ln(t)^{m-1}}{(m-1)!} + T_{2,0} \cdot \frac{\ln(t)^{m-2}}{(m-2)!} + \dots + T_{m,0}, \quad (2)
\end{gathered}
$$

etc.

In each of these ‘sets’ of formal solutions, the series (up to a $t^\alpha$ overall, $\alpha = 0, 1/2, ...$) are analytical at $t = 0$. One of the series $S_0, T_0, ...$ is the diagonal of the rational function annihilated by the (minimal order) differential operator $L_q$. It is therefore a globally bounded

<footnote>$^8$ We are thankful to Lairez for having clarified this point.</footnote>

<footnote>$^9$ The case where $Q_1$ and $Q_2$ depend on different sets of variables corresponds to a Hadamard product and will not be considered in this paper.</footnote>

<footnote>$^{10}$ This display of the formal series solutions as disentangled (1) and (2), ... solutions naturally provides a Jordan-block representation of the monodromy matrix at $t = 0$, where each block is an upper-triangular matrix.</footnote>
series. The other analytic solution series have, at first sight, no reason to be globally bounded series.

## 1.2. Conjecture on the number of variables

Our main conjecture corresponds to the *exact value* of the minimal number $N_v$ of variables of the rational functions required to represent a given diagonal of rational function series. Consider a rational function $R_1$ with $N$ variables, and the (minimal order) differential operator annihilating $\text{Diag}(R_1)$ (i.e. the diagonal of $R_1$), assumed to be homomorphic to its adjoint<fnref n="11" /> (i.e. its differential Galois group is included either in a symplectic, or in an orthogonal group). Among the formal solutions, at the origin, of this differential operator, there is one formal solution which has the *highest log-power*, $n$, i.e. behaves as $\ln(t)^n$. We conjecture that the diagonal of $R_1$ identifies with the diagonal of a rational function $R$ which depends on a minimal number of variables $N_v$, where $N_v$ is simply related to the *highest log exponent*<fnref n="12" /> $n$ by the following relation (with $N_v \le N$):

$$N_v = n + 2. \qquad (3)$$

Note that the existence of such rational functions with a minimal number of variables $N_v$ (*regardless of the number of monomials* and *degrees*), *does not prevent* the existence of other rational functions with more variables, giving the same diagonal. For instance, when the general term of the series writes as nested sums of binomials [1, 17], it is straightforward to obtain a first rational function [1]. This first rational function has often more variables than this minimal number $N_v$. Also note that a balanced ratio of factorials coefficients can be written in various binomial forms, thus yielding many rational functions.

## 1.3. Some remarks

Let us give some remarks on the definition of what we call the rational functions and on the number of variables occurring there.

We have seen in previous papers (see for instance section 4.3 and appendix G in [18]), that an exact result on the diagonal of a rational function of some variables $x, y, \dots$, depending on *parameters*, can straightforwardly be generalized to the same rational function, but where the *parameters* become arbitrary (rational or algebraic) *functions* of the product $t = xyz \dots$ of the variables of the rational function. Consequently we extend the definition of rational functions to rational functions where the variables are rescaled by arbitrary (rational or algebraic) functions of the product $t = xyz \dots$

In the sequel, some of the rational functions depend on $N$th root of variables (e.g. $u^{1/4}$, $v^{1/6}$, ...). The calculation of the diagonal of such a function, is equivalent to the calculation of the diagonal of this function where all the variables are raised to some power. In the following we will say, by abuse of language, in such cases, that we have a rational function *even if it contains Nth root of some variables*.

<footnote n="11">¹¹ It has been noticed in several of our papers [16, 19], that diagonal of rational (or algebraic) functions almost systematically yield differential operators which are *homomorphic to their adjoint*. The rare cases breaking this ‘self-adjoint duality’ were seen to correspond to candidates to rule out Christol’s conjecture [4, 15]. The question of this self-adjoint symmetry breaking is addressed in section 5.2 of [16].</footnote>

<footnote n="12">¹² The crucial role played by the highest log exponent corresponds to the concept of monodromy filtration (see paragraph 4.2 page 40 of [15]).</footnote>
Also, we have seen examples of rational functions where some monomials or variables in the rational function do not contribute to the diagonal, meaning that the rational function has in fact lesser number of variables. We used, for this situation, the term ‘effective’ number of variables in section 2.4 of [16]. Consider the rational function $1/Q$ where $Q = 1 - y - z - xz - xu - xzu - xyu$, which depends on four variables. The diagonal of $1/Q$ is ‘blind’$^{13}$ on the monomial $xz$ and the product $xu$ stands for one variable, reducing $Q$ to $\tilde{Q} = 1 - y - z - v - yv - zv$ which depends only on three variables. We should note that we have not seen this situation in our examples coming from physics or geometry.

## 1.4. Complements and additional speculations on conjecture (3)

Among the different sets (1) and (2), ... of formal series solutions of the (minimal order) differential operator, one set, for instance (1), corresponds to the highest power of the logarithms, namely $n$. We also conjecture that the series with no logarithm in this set, here $S_0$, will necessarily be a globally bounded series, and, thus, according to Christol’s conjecture [4], will be a diagonal of a rational function, with, according to the conjecture (3), the minimal number of variables $N_v = n + 2$. The other sets, for instance (2), will, in general, have a power of logarithms $m$ which is less than the highest power $n: m < n$. In this $m < n$ case we also conjecture that the power series (up to a $t^\alpha$ overall, $\alpha = 0, 1/2, \dots$) with no logarithm in this set, here $T_0$ for (2), cannot be a globally bounded series. Consequently, it cannot be a diagonal of a rational function.

In the cases where another set (2), is such that its power of logarithms, $m$, is actually equal to the highest power of logarithm $n$, we conjecture that the power series (up to a $t^\alpha$ overall, $\alpha = 0, 1/2, \dots$) with no logarithm in this set, here $T_0$, will necessarily also be a globally bounded series, and, thus, according to Christol’s conjecture [4], will be a diagonal of a rational function, with, according to the conjecture (3), the minimal number of variables $N_v = n + 2$.

Let us recall that the definition of diagonal of rational functions is based on multi-Taylor expansions around the origin [1]. Therefore, the seeking of the maximum exponent of the logarithm in the formal solutions is also made at the origin. The maximum exponent of the logarithms in the formal solutions around the other singularities can also be considered, and this is checked for many of our examples below.

## 1.5. Conjecture on the parity of the number of variables and the differential Galois groups

Another conjecture is related to the symplectic, or orthogonal, character of the differential Galois group$^{14}$ of the irreducible differential operator annihilating the diagonal of the rational function with $N_v$ variables.

Since we assume that the (minimal order) irreducible differential operator annihilating the diagonal of the rational function is *homomorphic to its adjoint*, its differential Galois group is necessarily symplectic or orthogonal [19].

We forward a conjecture stating that the parity of the ‘minimal’ number $N_v$ of variables in the irreducible denominator $Q$ dictates the character either symplectic $Sp(N_v)$ is an odd number) or orthogonal $SO(N_v)$ is an even number) of the differential Galois group of the (minimal

<footnote>$^{13}$ Changing the monomial $xz$ into $\mu xz$, the resulting diagonal will not depend on the parameter $\mu$.</footnote>

<footnote>$^{14}$ An irreducible differential operator $L_q$, of order $q$, has, generically, a symmetric square ($\mathrm{sym}^2(L_q)$) of order $N_s = q(q+1)/2$ and an exterior square ($\mathrm{ext}^2(L_q)$) of order $N_e = q(q-1)/2$. If $\mathrm{sym}^2(L_q)$ (resp. $\mathrm{ext}^2(L_q)$) annihilates a rational solution, or is of order $N_s - 1$ (resp. $N_e - 1$), the differential operator $L_q$ is included in the *orthogonal group* $SO(q, \mathbb{C})$ (resp. *symplectic group* $Sp(q, \mathbb{C})$) that admits an invariant quadratic (resp. alternating) form.</footnote>
order) differential operator annihilating the diagonal of the rational function. This conjecture is that the group is orthogonal for $N_v$ even and symplectic for $N_v$ odd:

$$N_v \text{ even (resp. odd)} \rightarrow SO (\text{resp. Sp}) \qquad (4)$$

The illustrative examples displayed in this paper in favor of these two conjectures (3) and (4), are chosen for pedagogical reasons, but also for their interest *per se*. We have two different kinds of examples: the ones corresponding to (irreducible) differential operators which have *Maximum Unipotent Monodromy*<fnref n="15" /> (MUM), and the ones that do not have MUM. In the MUM case, we have a simple relation $n = q - 1$ between the power of logarithm $n$ in (3), and the order $q$ of the differential operator $L_q$. The conjecture (3) thus becomes, in the MUM case, a simple relation $N_v = q + 1$ between the number of variables and the order of the operator. For the non-MUM examples, the number of variables is *not* related to the order of the differential operators, but to the highest power of the logarithms $n$, in other words, *it is related to the monodromy matrix at the origin*.

The paper is organized as follows. We recall in section 2 the definition of the diagonal of a rational function, and in section 3 we recall the results of the LGFs of the $d$-dimensional (f.c.c, s.c and diamond) lattices, to illustrate the two conjectures (3) and (4). Section 4 presents some illustrative examples for non factorizable multivariate polynomials $Q$. We give, in section 5, the polynomials $Q$ for some Calabi-Yau equations [10] of geometric origin, obtaining, for each case, the polynomial $Q$ with the minimum number of variables $N_v = 3 + 2 = 5$. Section 6 deals with the situation where the denominator polynomial $Q$ factorizes as $Q = Q_1 Q_2$, giving either a direct sum, or a unique factorization, of the (minimal order) differential operator annihilating the diagonal of $1/Q$. In section 7, we give examples of diagonal of algebraic functions which are $N$th root of rational functions and equivalent rational functions of the form $1/Q$, giving the same diagonals. Finally, in section 8, we discuss the homomorphisms-to-adjoint assumption on a counter-example candidate to Christol's conjecture [4].

## 2. Diagonals and multinomial expansion

The diagonal of a rational function $R$, dependent on (for example) three variables, is obtained by (multi-Taylor) expanding $R$ around the origin

$$R(x,y,z) = \sum_m \sum_n \sum_l a_{m,n,l} \cdot x^m y^n z^l, \qquad (5)$$

keeping only the terms such that $m = n = l$. The diagonal reads, with $t = xyz$:

$$\mathrm{Diag}(R(x,y,z)) = \sum_m a_{m,m,m} \cdot l^m. \qquad (6)$$

In most of the examples of this paper the rational function will have the form $1/Q$ with

$$Q = 1 - (T_1 + T_2 + \cdots + T_n), \qquad (7)$$

where the $T_j$'s are monomials. The expansion of the rational function $1/Q$ reads:

$$\frac{1}{Q} = \sum \frac{(k_1 + k_2 + \cdots + k_n)!}{k_1! k_2! \cdots k_n!} \cdot T_1^{k_1} T_2^{k_2} \cdots T_n^{k_n}. \qquad (8)$$

<footnote n="15">¹⁵ Maximum unipotent monodromy: the critical exponents at the origin are all equal.</footnote>
Calculating the diagonal of $1/Q$ amounts to distributing the powers $k_j$ on the variables occurring in the monomials, and putting equal the exponents of each variable.

With the example<fnref n="16" /> $Q = 1 - x - y - z - u^{1/2} - v^{1/5}$, depending on five variables, or nth root of variables, the expansion of $1/Q$, around the origin, reads:

$$ \frac{1}{Q} = \sum_{k_i=0, i=1, \dots, 5}^{\infty} \frac{(k_1 + k_2 + k_3 + k_4 + k_5)!}{k_1! k_2! k_3! k_4! k_5!} \cdot x^{k_1} y^{k_2} z^{2k_3} u^{k_4/2} v^{k_5/5}. \quad (9) $$

The diagonal will extract the terms with the same power. Introducing the integer $p$, such that $k_1 = k_2 = k_3 = k_4/2 = k_5/5 = p$, the only terms contributing to the diagonal are

$$ \sum_{p=0}^{\infty} \frac{(p+p+p+2p+5p)!}{p! p! p! (2p)! (5p)!} \cdot x^p y^p z^p u^p v^p = \sum_{p=0}^{\infty} \frac{(10p)!}{p!^3 (2p)! (5p)!} \cdot t^p. \quad (10) $$

Let us mention how we obtain the (minimal order) differential operator annihilating the diagonal. Once a long series is obtained, we use the guessing method to obtain<fnref n="17" /> the differential equation (ODE). We make use of the ‘ODE formula’ forwarded in section 3.1 of [21] (and with details in section 1.2 of [12]) to ensure that we actually deal with the minimal order differential equation. For the differential operators of high order, which are not irreducible, the factorization is obtained using the method of factorization of differential operators modulo primes (see section 4 in [22] and remark 6 in [19]).

### 3. The LGF of the s.c, diamond and f.c.c lattices

For the LGFs of the s.c lattice of dimension $d$, the rational function have $N_v = d + 1$ variables. The corresponding differential operators $L_n$, ($n = d$, see appendix A.1.1) annihilating the LGF have MUM, and thus, the formal solution with the highest log-power at the origin corresponds to $\ln^{n-1}(t)$. The relation of conjecture (3) is satisfied.

Similarly, one may consider the LGF of the diamond lattice (appendix A.1.2) in 3, 4 and 5 dimensions, where the rational functions depend on respectively 4, 5 and 6 variables, and check that the formal solutions with the highest log-power, at the origin, are respectively in $\ln(t)^2$, $\ln(t)^3$ and $\ln(t)^4$, in agreement with conjecture (3). All these differential operators also have MUM.

The LGFs of the f.c.c lattice of dimension $d$ are diagonals of rational functions of the form $1/Q$ for the dimension $d < 7$, see [5–8], and we have produced the corresponding (minimal order) differential operators [9] for $d = 7, 8, \dots, 12$. These differential operators no longer<fnref n="18" /> have MUM. According to the parity of the dimension $d$, which is related to the number of variables by $N_v = d + 1$, the differential Galois group of these differential operators are included in $Sp(N_v$ odd) or $SO(N_v$ even) in agreement with our conjecture (4). We have checked (see appendix A.2) that, among the formal solutions at the origin of these differential operators, one solution is in front of $\ln(t)^n$, where $n$ is the highest log-power. This exponent is in

<footnote n="16">¹⁶ For ‘polynomials’ with nth root of variables, see, for instance, (33) in section 5.</footnote>

<footnote n="17">¹⁷ Alternately, we can use [16] the creative telescoping [20] to get this differential operator as a telescoper. This method often requires more computing time.</footnote>

<footnote n="18">¹⁸ Except for $d = 2, 3, 4$ see table A.1 in appendix A.</footnote>
agreement with the relation $N_v = n + 2$ of the conjecture (3). For the exponent of the logarithmic formal solutions at all the other singularities see appendix A.2.

Before we consider some examples with irreducible (non factorizable) denominator $Q$, let us address the following comment.

We considered (assumed) the number of variables, occurring in the LGFs, as the minimal one, since the number of variables (when the LGF is written as a diagonal of a rational function) is related to the dimension of the lattice (when the LGF is written as multidimensional integral). We recall that our conjectures deal with diagonals of rational functions (written as the reciprocal of a multivariate polynomial). A question may arise as whether a LGF with $N$ variables could also be represented as the diagonal of a rational function with less variables (except for $N = 2$ or $N = 3$ where one is sure to have the smallest number of variables). To get a smaller number of variables, that could contradict our conjecture $N_v = n + 2$, requires that, starting from the multiple integral of rational function defining a given LGF, an integration on one variable, for instance, would give a result that can also be cast as a multiple integral of a rational function. Such a scenario seems highly unlikely<fnref n="19" />.

# 4. Examples with non factorizable denominator $Q$

## 4.1. A LGF-like non-MUM example

In the examples with the LGFs, one may imagine that the occurrence of the differential Galois groups $Sp$ or $SO$ is related to the dimension of the lattice. Let us go beyond this relation by considering the structure function

$$ \lambda = c_1 c_2 c_3 + c_1 c_2 c_4 + c_1 c_3 c_4 + c_2 c_3 c_4, \quad c_j = \cos \phi_j, \quad (11) $$

considered by Guttmann in [6], which does not correspond to any known obvious lattice. The (minimal order) differential operator annihilating

$$ G(t) = \frac{1}{(2\pi)^4} \int \frac{d\phi_1 d\phi_2 d\phi_3 d\phi_4}{1 - t \cdot \lambda}, \quad (12) $$

is of order 8 and is irreducible. Its exterior square is of order 27, instead of the generic order $28 = (8 \times 7)/2$. Its differential Galois group is thus included in the symplectic group $Sp(8, \mathbb{C})$. The LGF $G(t)$ is the diagonal of the rational function $1/Q$, depending on 5 variables $(z_0, z_1, z_2, z_3, z_4)$

$$ Q = 1 - t \cdot \lambda, \quad (13) $$

such that $c_j = z_j + 1/z_j, j = 1, \dots, 4$ and $t = z_0 z_1 z_2 z_3 z_4$.

The eight formal solutions, at the origin, of the order-eight differential operator read

<footnote n="19">¹⁹ In contrast the diagonal of a rational function $1/Q_{eq}$ can be equal to the diagonal of a square root of a rational function $1/Q^{1/2}$, with one less variable, see (79) below. A good illustration corresponds to the known relation [6, 36, 37] between the LGF of the $d$-dim diamond and the LGF of the $(d+1)$-dim s.c lattices, see relation (73) below.</footnote>
$$
\begin{align}
& S^{(0)}, \qquad S^{(0)} \cdot \ln(t) + S_{2,0}, \qquad S^{(0)} \cdot \frac{\ln(t)^2}{2!} + S_{3,1} \cdot \ln(t) + S_{3,0}, \nonumber \\
& S^{(0)} \cdot \frac{\ln(t)^3}{3!} + S_{4,2} \cdot \frac{\ln(t)^2}{2!} + S_{4,1} \cdot \ln(t) + S_{4,0}, \nonumber \\
& t^{1/2} \cdot S^{(1/2)} \qquad t^{1/2} \cdot S^{(1/2)} \cdot \ln(t) + t^{1/2} \cdot S_{6,0}, \nonumber \\
& t^{1/3} \cdot S^{(1/3)}, \qquad t^{2/3} \cdot S^{(2/3)}, \tag{14}
\end{align}
$$

where the $S_{i,j}$'s are analytical series at $t=0$, and where the other series begin as $1+\cdots$. The series $S^{(0)}$ is globally bounded, and corresponds to the diagonal of the rational function. All the other analytical series $S^{(1/2)}$, $S^{(1/3)}$, $S^{(2/3)}$, $S_{i,j}$ in (14) are not globally bounded.

The diagonal $S^{(0)} = \text{Diag}(1/Q)$ is the diagonal of a rational function of $N_v = 3+2=5$ variables, in agreement with conjecture (3).

Around all the other singularities, the highest log-power is in $\ln(t)^1$, and each formal series, in front of $\ln(t)^1$, is not globally bounded.

The generalization of (11) to 6 variables amounts to considering:

$$ \lambda = c_1 c_2 c_3 c_4 + c_1 c_2 c_3 c_5 + c_1 c_2 c_4 c_5 + c_2 c_3 c_4 c_5, \quad c_j = \cos \phi_j. \quad (15) $$

The corresponding differential operator is expected to have its differential Galois group included in the orthogonal group. The differential operator is of order 9. Its differential Galois group is indeed (included) in $SO(9, \mathbb{C})$, (its symmetric square being of order 44, instead of the generic order $45 = (9 \times 10)/2$).

The formal solution of this order-nine differential operator with the highest log-power at the origin, corresponds to $\ln(t)^4$, giving, according to the conjecture (3), $N_v = 4+2=6$ as the minimal number of variables occurring in the rational function.

## 4.2. 3D fcc example: reduction to three variables

The diagonal of $1/Q$ where

$$ Q = 1 - xyz \cdot \left( \left( x + \frac{1}{x} \right) \cdot \left( y + \frac{1}{y} \right) + \left( x + \frac{1}{x} \right) \cdot \left( z + \frac{1}{z} \right) + \left( y + \frac{1}{y} \right) \cdot \left( z + \frac{1}{z} \right) \right), \quad (16) $$

is a polynomial depending on four variables $x, y, z, u$, reproduces the three-dimensional f.c.c LGF. The (minimal order) differential operator annihilating this diagonal is of order three, and its differential Galois group is included in $SO(3, \mathbb{C})$. The most singular formal solution is in $\ln(t)^2$, in agreement with 2+2 variables for the rational function, in agreement with conjecture (3).

Let us reduce the number of variables of the polynomial $Q$, given in (16), to three variables, by fixing $u=1$. The polynomial $Q$ becomes:

$$ Q_1 = 1 - xyz \cdot \left( \left( x + \frac{1}{x} \right) \cdot \left( y + \frac{1}{y} \right) + \left( x + \frac{1}{x} \right) \cdot \left( z + \frac{1}{z} \right) + \left( y + \frac{1}{y} \right) \cdot \left( z + \frac{1}{z} \right) \right). \quad (17) $$

The (minimal order) differential operator, annihilating this diagonal, is of order six, and its differential Galois group is included in $Sp(6, \mathbb{C})$. The formal solution with the highest log-power at the origin is in $\ln(t)^1$, in agreement with 1+2=3 variables for the rational function.
### 4.3. Another diagonal representation of the 3D fcc LGF

The LGF of the three-dimensional fcc lattice can also be seen as the diagonal of $1/Q$, where the polynomial denominator $Q$ depends on four variables:

$$Q = 1 - x^2 yzu \cdot (1 + 4xyzu) - (1 + u) \cdot (y + z). \qquad (18)$$

The diagonal of $1/Q$, where $Q$ depends on four variables, gives (with $t = xyzu$):

$$\mathrm{Diag}_{(xyzu)}\left(\frac{1}{Q}\right) = {}_3F_2\left(\left[\frac{1}{2}, \frac{1}{3}, \frac{2}{3}\right], [1, 1], 108 \cdot t^2 \cdot (1 + 4t)\right). \qquad (19)$$

The formal solution of the corresponding differential operator, with the highest log-power at the origin, is in $\ln(t)^2$, in agreement with the minimal number of variables of $4 = 2 + 2$, according to conjecture (3).

Taking the diagonal of $1/Q$ with $Q$ given in (18) on only the 3 variables $(x, y, z)$, which means that the variable $u$ is seen as a parameter, one obtains (with $s = xyz$):

$$\mathrm{Diag}_{(xyz)}\left(\frac{1}{Q}\right) = {}_2F_1\left(\left[\frac{1}{3}, \frac{2}{3}\right], [1], 27 \cdot u \cdot (1+u)^2 \cdot (1+4us) \cdot s^2\right). \qquad (20)$$

The formal solution of the corresponding differential operator with the highest log-power at the $s = 0$ origin, is in $\ln(s)^1$, in agreement with $3 = 1 + 2$ variables, according to conjecture (3).

Here, for the same rational function $1/Q$, we clearly see that the fact that the diagonal is taken on 4 or 3 variables, produces a singular formal solution behaving in $\ln(t)^2$ or $\ln(s)$.

### 4.4. A reflexive polytope example

From the 210 differential operators arising from reflexive four-polytopes [17, 23] (Laurent polynomials of four variables) with symplectic differential Galois group $Sp$, let us consider the example$^{20}$, depending on five variables, which corresponds to the rational function $1/Q$ with polynomial denominator

$$Q = 1 - xyzuv \cdot S, \qquad (21)$$

where:

$$S = x + y + z + u + \frac{1}{x} + \frac{1}{y} + \frac{1}{z} + \frac{1}{u} + yz + \frac{x}{z} + \frac{u}{z} + \frac{x}{yz} + \frac{z}{xu} + \frac{u}{yz} + \frac{xu}{yz} + \frac{yz}{x} + \frac{yz}{u} + \frac{xu}{z^2y}. \qquad (22)$$

The diagonal of $1/Q$ is annihilated by an irreducible order-six differential operator whose differential Galois group is symplectic (included in $Sp(6, \mathbb{C})$). The formal solution, at the origin, of this order-six differential operator, with the highest log-power, behaves as $\ln(t)^3$. This is in agreement with the number of variables $N_v = 3 + 2 = 5$ occurring in the polynomial $Q$, according to conjecture (3).

Around all the other singularities, the maximum power of the log in the formal solutions is 1, and the series are non-globally bounded.

<footnote>$^{20}$ Polytope v18.10 805, topology 28 in [17, 23].</footnote>
### 4.5. An Apery generalization example

Let us recall the series with Apery numbers<fnref n="21" />:

$$ \sum_{n=0}^{\infty} \sum_{k=0}^{n} \binom{n}{k}^2 \binom{n+k}{k}^2 \cdot t^n = \sum_{n=0}^{\infty} \sum_{k=0}^{n} \left( \frac{(n+k)!}{k!^2 (n-k)!} \right)^2 \cdot t^n. \quad (23) $$

This series is actually the diagonal of the rational function 1/Q with four variables [15]:

$$ Q = (1 - x_1 - x_2) \cdot (1 - x_3 - x_4) - x_1 x_2 x_3 x_4. \quad (24) $$

More generally, let us consider the series given by (with m a positive integer):

$$ \sum_{n=0}^{\infty} \sum_{k=0}^{n} \binom{n}{k}^m \binom{n+k}{k}^m \cdot t^n = \sum_{n=0}^{\infty} \sum_{k=0}^{n} \left( \frac{(n+k)!}{k!^2 (n-k)!} \right)^m \cdot t^n. \quad (25) $$

This series is the diagonal of the rational function 1/Qm, with<fnref n="22" />

$$ Q_m = 1 - \prod_{j=0}^{m-1} (x_{2j+1} + x_{2j+2} + x_{2j+1} \cdot x_{2j+2}), \quad (26) $$

depending on 2m variables. For any value of m, the number of variables being even, the differential operator, corresponding to the diagonal of 1/Qm, has a differential Galois group which should be included in the orthogonal group SO, according to conjecture (4).

For m = 3, we have six variables in (26),

$$ Q_3 = 1 - (x_1 + x_2 + x_1 x_2) \cdot (x_3 + x_4 + x_3 x_4) \cdot (x_5 + x_6 + x_5 x_6), \quad (27) $$

and the diagonal of the rational function 1/Q3 reads:

$$ \begin{aligned} \sum_{n=0}^{\infty} \sum_{k=0}^{n} \binom{n}{k}^3 \binom{n+k}{k}^3 \cdot t^n &= \sum_{n=0}^{\infty} \sum_{k=0}^{n} \left( \frac{(n+k)!}{k!^2 (n-k)!} \right)^3 \cdot t^n \\ &= 1 + 9t + 433t^2 + 36729t^3 + \dots \end{aligned} \quad (28) $$

This series (28) is annihilated by an order-nine differential operator L9. The symmetric square of L9 is of order 44, instead of the generic order $45 = (9 \times 10)/2$. The differential Galois group of this order-nine differential operator L9 is (included) in the orthogonal group SO(9, C). The formal solution of L9 with the highest log-power, at the origin, behaves as $\ln(t)^4$, in agreement with the conjecture (3): $N_v = 4 + 2 = 6$.

The maximum exponent of the log's in the formal solutions around all the other singularities $t = t_j$ (with the exception of $t = \infty$) is 2, and the series, in front of $\ln(t - t_j)^2$, are not globally bounded.

For the singularity $t = \infty = 1/s$, the highest log-power is also 4, i.e. the same value as the maximum log-power around $t = 0$, so we expect the corresponding series in front of $\ln(s)^4$ to be globally bounded. The first terms of the series read

$$ s \cdot (1 + 9s + 433s^2 + 36729s^3 + \dots). \quad (29) $$

<footnote n="21">21 The sequence is related to Apery's proof on the irrationality of $\zeta(3)$.</footnote>

<footnote n="22">22 Use the multinomial expansion, then equate the exponents of the variables.</footnote>
This series (29) identifies with (28) up to a multiplicative variable s, it is therefore globally bounded. This relation is a consequence of the symmetry $t \to 1/t$ in the differential operator $L_9$

$$L_9\left(\frac{1}{t}\right) = t \cdot L_9(t) \cdot \frac{1}{t}. \qquad (30)$$

For the rational function $1/Q_3$, the change $x_j \to 1/x_j$ appears as

$$\text{Diag}\left(\frac{t}{Q_3}\right) = \text{Diag}\left(-\frac{1}{\tilde{Q}_3}\right) \quad \text{with:} \\ \tilde{Q}_3 = 1 - \frac{1}{t} \cdot (1+x_1+x_2) \cdot (1+x_3+x_4) \cdot (1+x_5+x_6). \qquad (31)$$

Similarly, for $m = 4$, we have eight variables in (26). The differential operator, annihilating the diagonal of the rational function $1/Q_4$, is of order 15. Its differential Galois group is (included in) the orthogonal group $SO(15, \mathbb{C})$. The formal series solution, at the origin, of the order fifteen differential operator, with the highest log-power, behaves as $\ln(t)^6$ in agreement with the conjecture (3): $N_v = 6 + 2 = 8$.

## 5. Some Calabi-Yau examples

We consider, in this section, some examples of Calabi-Yau equations from [10], where the expressions of the general term of the series are known in closed form. These order-four differential operators are irreducible, their differential Galois groups are (included in) symplectic groups $Sp(4, \mathbb{C})$. They all have the MUM property [10], and their formal solutions with the highest log-power behave as $\ln(t)^3$. Therefore, according to conjecture (3), these differential operators should annihilate the diagonal of a rational function of $N_v = 3 + 2 = 5$ variables.

### 5.1. The first 19 Calabi-Yau operators in Almkvistet al table [10]

We have considered the first 19 Calabi-Yau differential operators of [10]. These Calabi-Yau equations (except #9) have a geometric origin [24–28]. The general term of the series are (or can be) written as nested sums of products of binomials [17, 29], known to correspond to diagonals of rational functions [1, 17, 29]. The aim, here, is to give the rational function with only five variables.

Assuming some $1/Q$ form for the rational function, the polynomials $Q$ are obtained from closed formulae (given in [10]) of the general term written as ratio of factorials, using a ‘guessing’ procedure sketched in appendix B. The procedure in appendix B, amounts, from the general term given in [10], to going back up to the expression coming out from a multinomial expansion. Eventually one finds that all the polynomials $Q$ can be written with five variables. We give the corresponding multivariate polynomials $Q$ obtained from this guessing procedure, in table 1.

Note that, for some $Q$'s, we still say that $Q$ is ‘polynomial’ even if it contains $N$th root of some variables ($u^{1/4}$, $v^{1/6}$, ...). Section 2 shows that the diagonal of the rational function with $N$th root of variables is actually a power series (not a Puiseux series). In the following, instead of some ‘polynomials’ containing $N$th root of variables, we give equivalent polynomials still with five variables.
<table><caption>**Table 1.** Rational functions $1/Q$ for some Calabi–Yau series $\sum A_n t^n$, where $A_n$ is the general term of the series given in [10]. We follow the numbering # of [10].</caption><thead><tr><th># A<sub>n</sub></th><th>1-Q</th><th>N<sub>v</sub></th></tr></thead><tbody><tr><td>1 <sup>(5n)!</sup>/<sub>n!5</sub></td><td>x + y + z + u + v</td><td>5</td></tr><tr><td>2 <sup>(10n)!</sup>/<sub>n!3(2n)!(5n)!</sub></td><td>x + y + z + u<sup>1/2</sup> + v<sup>1/5</sup></td><td>5</td></tr><tr><td>3 <sup>(2n)!4</sup>/<sub>n!8</sub></td><td>(x + z)(1 + y)(1 + v)(1 + u)</td><td>5</td></tr><tr><td>4 <sup>(3n)!</sup>/<sub>n!3</sub><sup>2</sup></td><td>(x + yu + zv)(y + xz + uv)</td><td>5</td></tr><tr><td>5 <sup>(2n)!2(3n)!</sup>/<sub>n!7</sub></td><td>x + y + z + u + v(x + y)(z + u)</td><td>5</td></tr><tr><td>6 <sup>(2n)!4n!</sup>/<sub>n!6</sub></td><td>x + y + z + u + v(z + u)</td><td>5</td></tr><tr><td>7 <sup>(8n)!</sup>/<sub>n!4(4n)!</sub></td><td>x + y + z + u + v<sup>1/4</sup></td><td>5</td></tr><tr><td>8 <sup>(6n)!</sup>/<sub>n!4(2n)!</sub></td><td>x + y + z + u + v<sup>1/2</sup></td><td>5</td></tr><tr><td>9 <sup>(2n)!(12n)!</sup>/<sub>n!4(4n)!(6n)!</sub></td><td>x + y + z(x + y) + u<sup>1/4</sup> + v<sup>1/6</sup></td><td>5</td></tr><tr><td>10 <sup>(4n)!</sup>/<sub>n!5(2n)!</sub><sup>2</sup></td><td>(x + yu + (zv)<sup>1/2</sup>)(y + xz + (uv)<sup>1/2</sup>)</td><td>5</td></tr><tr><td>11 <sup>(3n)!(4n)!</sup>/<sub>n!5(2n)!</sub></td><td>x + y + z + u + v(y + z + u)</td><td>5</td></tr><tr><td>12 <sup>(4n)!(6n)!</sup>/<sub>n!3(2n)!2(3n)!</sub></td><td>x + y + (1 + z)(u<sup>1/2</sup> + v<sup>1/2</sup>)</td><td>5</td></tr><tr><td>13 <sup>(6n)!</sup>/<sub>n!(2n)!(3n)!</sub><sup>2</sup></td><td>(x + (yu)<sup>1/2</sup> + (zv)<sup>1/3</sup>)(y + (xz)<sup>1/2</sup> + (uv)<sup>1/3</sup>)</td><td>5</td></tr><tr><td>14 <sup>(2n)!(6n)!</sup>/<sub>n!5(3n)!</sub></td><td>x + y + z + u(x + y) + v<sup>1/3</sup></td><td>5</td></tr><tr><td>15 &sum;<sub>k=0</sub><sup>n</sup> <sup>(3n)!</sup>/<sub>n!3</sub> <sup>(n k)</sup> <sup>3</sup></td><td>x + y + z(x + v) + u(y + v)</td><td>5</td></tr><tr><td>16 &sum;<sub>k=0</sub><sup>n</sup> <sup>(2n)</sup>/<sub>n</sub> <sup>(n k)</sup> <sup>2</sup> <sup>(2k)</sup>/<sub>k</sub> <sup>(2n - 2k)</sup>/<sub>n - k</sub></td><td>x + y + z + u + v(xyz + xyu + xzu + yzu)</td><td>5</td></tr><tr><td>17 &sum;<sub>k=0</sub><sup>n</sup> &sum;<sub>j=0</sub><sup>n-k</sup> <sup>n!</sup>/<sub>j!k!(n-j-k)!</sub><sup>3</sup></td><td>(x + y + z)(v + xu + yz)(uv + zv + xyu)</td><td>5</td></tr><tr><td>18 &sum;<sub>k=0</sub><sup>n</sup> <sup>(2n)</sup>/<sub>n</sub> <sup>(n k)</sup> <sup>4</sup></td><td>u + (x + y)(x + v)(z + v)(y + z)</td><td>5</td></tr><tr><td>19 &sum;<sub>k=0</sub><sup>n</sup> <sup>(n k)</sup> <sup>3</sup> <sup>(n + k)</sup>/<sub>n</sub> <sup>(2n - k)</sup>/<sub>n</sub></td><td>(x + y)(u + x)(u + v)(y + z)(v + z + y + u)</td><td>5</td></tr></tbody></table>

## 5.2. The Q's are multivariate polynomials of variables and Nth root of variables

For some Q's which are polynomials with N-roots of variables, a straightforward change of variables, such $(x, y, \dots) \to (x^n, y^n, \dots)$, may be introduced.

For instance, for Calabi–Yau number 2 (that we denote $CY_2$),

$$ 
\begin{aligned}
CY_2 &= \sum_{n=0}^{\infty} \frac{(10n)!}{n!^3 (2n)!(5n)!} \cdot t^n = {}_4F_3 \left( \left[ \frac{1}{10}, \frac{3}{10}, \frac{7}{10}, \frac{9}{10} \right], [1, 1, 1], 2^8 5^5 \cdot t \right) \\
&= 1 + 15120t + 3491888400t^2 + 1304290155168000t^3 + \dots
\end{aligned} 
\quad (32) $$

which is actually the diagonal of $1/Q$, with

$$ Q = 1 - (x + y + z + u^{1/2} + v^{1/5}), \quad (33) $$
the series is in $t = xyzuv$. If one wants to get rid of the Nth roots $u^{1/2}$ and $v^{1/5}$, one can rather consider the polynomial

$$Q = 1 - (x^{10} + y^{10} + z^{10} + u^{5} + v^{2}), \qquad (34)$$

yielding for the diagonal of $1/Q$ where $Q$ is given by (34), the series (32) where $t$ is changed into $t^{10}$. Other polynomials with five variables can be found. With the polynomial

$$Q = 1 - (x + y + z + x^4 y^4 u^4 v^5 + xzvu^2), \qquad (35)$$

or the other polynomial

$$Q = 1 - (xyzu + xzuv + yzuv + xyvu^2 + xyzv^3), \qquad (36)$$

the series (32) is reproduced with $t$ changed respectively into $t^6$ and $t^9$.

### 5.3. The rational function versus the pullbacked solution

From the Calabi–Yau equations in [10], two (or more) may have the same Yukawa coupling$^{23}$, which means [31] that one solution can be written in terms of the other one. Let us see how this property appears in the multivariate rational functions $1/Q$.

#### 5.3.1. Calabi–Yau number 79. The series for Calabi–Yau number 79 reads

$$ \begin{aligned} CY_{79} &= \sum_{n=0}^{\infty} \sum_{k=0}^{n} \binom{n}{k} \cdot \frac{(5k)!}{k!^5} \cdot t^n \\ &= 1 + 121t + 113641t^2 + 168508561t^3 + \dots \end{aligned} \qquad (37) $$

and has the same Yukawa coupling [10] as $CY_1$. It reads

$$ \begin{aligned} CY_{79} &= \frac{1}{1-t} \cdot CY_1 \left( \frac{t}{1-t} \right) \\ &= \frac{1}{1-t} \cdot {}_4F_3 \left( \left[ \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} \right], [1, 1, 1], 5^5 \cdot \frac{t}{1-t} \right), \end{aligned} \qquad (38) $$

which is the diagonal of $1/Q_{79}$ where $Q_{79}$ reads (with $t = xyzuv$):

$$Q_{79} = (1-t) \cdot \left( 1 - \left( \frac{x}{1-t} + y + z + u + v \right) \right). \qquad (39)$$

The diagonal of $1/Q$, with $Q$ given in (39), is the same series as in (37).

<footnote>$^{23}$ See, e.g. [30–32] for the definition of the Yukawa coupling, and appendix G.</footnote>
5.3.2. Calabi–Yau number 128. Another example is Calabi–Yau number 128 whose series reads

$$ 
\begin{aligned}
CY_{128} &= \sum_{n=0}^{\infty} \sum_{k=0}^{n} \binom{2n}{n} \binom{n}{k} \frac{(5k)!}{k!^3 (2k)!} t^n \\
&= 1 + 122t + 114126t^2 + 169305620t^3 + 307902541870t^4 + \dots
\end{aligned} 
\quad (40) $$

and has the same Yukawa coupling [10] as $CY_1$, in terms of which it writes:

$$ 
\begin{aligned}
CY_{128} &= \frac{1}{\sqrt{1-4t}} \cdot CY_1 \left( \frac{t}{1-4t} \right) \\
&= \frac{1}{\sqrt{1-4t}} \cdot 4F_3 \left( \left[ \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} \right], [1, 1, 1], 5^5 \cdot \frac{t}{1-4t} \right).
\end{aligned} 
\quad (41) $$

The Calabi–Yau series (40) or (41), is the diagonal of $1/Q_{128}$, where the denominator $Q_{128}$ has an ‘algebraic term’ depending only on the variable $t = xyzuv$:

$$ Q_{128} = \sqrt{1-4t} \cdot (1 - (z+u+v)) - x - y. \quad (42) $$

## 6. Factorizable denominator $Q$: examples with two factors

### 6.1. The differential operators occurring in the $\chi^{(n)}$'s of the Ising model

The rational functions, considered in the previous sections, are in the form $1/Q$, where the denominator $Q$ is a non factorizable multivariate polynomial, the (minimal order) differential operator annihilating these diagonals, being irreducible, contrary to the differential operators annihilating integrals corresponding to the $n$-fold integrals $\chi^{(n)}$'s of the susceptibility of the Ising model. These integrals are very convoluted forms of algebraic fractions, which have been shown to be diagonal of rational functions [1], but are far from being in the form $1/Q$.

All the factors, occurring in the differential operators for the $\chi^{(n)}$'s, have been shown to have differential Galois groups either symplectic or orthogonal (see [19] and references therein). Furthermore, focusing on the blocks of factors (occurring in the differential operators corresponding to the $\chi^{(n)}$'s) which have a unique factorization, i.e. that write as, (e.g. for three factors) $L_n L_p L_q$, it appears that if $L_q$ is in the orthogonal group (resp. symplectic group), the left factor $L_p$ is in the symplectic group (resp. orthogonal group) and so on, in an alternating way. Appendix C gives the situation for all the factors occurring in the differential operators corresponding to $\chi^{(3)}, \dots, \chi^{(6)}$.

This occurrence of products, as well as direct sums of products, for factors of the differential operators annihilating these diagonals, seems to be related to the fact that the denominator $Q$ is not irreducible. In the sequel, we address this case, restricting for pedagogical reason, to only two factors. We will see, in these examples with two factors, that conjecture (3) is still valid. The fact that the denominator $Q$ is not irreducible yields differential operators that are not irreducible: thus one cannot simply introduce a differential Galois group for these differential operators. One has to consider the differential Galois group of each factor of these reducible differential operators. We will see that an alternating symplectic/orthogonal structure seems to systematically occur (as appendix C shows for the $\chi^{(n)}$'s).
## 6.2. Foreword: two factors

In all the examples of the form $1/Q = 1/Q_1/Q_2$ displayed below, and many more not given in this paper, we have systematically obtained the following results. If the number of variables, and the variables for $Q_1$ and $Q_2$ are the same, the differential operator, annihilating the diagonal of $1/Q = 1/Q_1/Q_2$, is of the form

$$ (\mathcal{L}^{(1)} \oplus \mathcal{L}^{(2)}) \cdot N, \qquad (43) $$

where the differential operators $\mathcal{L}^{(1)}$ and $\mathcal{L}^{(2)}$ are homomorphic to the differential operators annihilating, respectively, the diagonal of $1/Q_1$ and $1/Q_2$, and where the differential operator $N$, at the right, is some ‘dressing’. We have not found any simple interpretation of this ‘dressing’.

In contrast, if the number of variables for $Q_1$ and $Q_2$ is different, the variables for (for instance) $Q_2$ being a subset of the set of variables for $Q_1$, the differential operator, annihilating the diagonal of $1/Q = 1/Q_1/Q_2$, is of the form

$$ \mathcal{L}^{(1)} \cdot N, \qquad (44) $$

where the differential operator $\mathcal{L}^{(1)}$ is homomorphic to the differential operator annihilating the diagonal of $1/Q_1$, and where, again, we have not found any simple interpretation$^{24}$ of this ‘dressing’ differential operator $N^{25}$.

## 6.3. First example

As a first example, consider the diagonal of $1/Q$

$$ Q = 1 - x - y - z + y^2 - z^2 + xy + yz + yz^2 + xz^2 + z^3, \qquad (45) $$

which is annihilated by an irreducible order-ten differential operator $L_{10}$ which has a differential Galois group (included) in $Sp(10, \mathbb{C})$. The formal solution with the highest log-power at the origin is in $\ln(t)^1$, in agreement with $3 = 1 + 2$ variables.

Changing the monomial $-y$ into $-2y$ in (45), one obtains a polynomial $\tilde{Q}$ that, now, factorizes:

$$ \tilde{Q} = (1 - (x + y + z)) \cdot (1 - y - z^2). \qquad (46) $$

The diagonal of $1/\tilde{Q}$ is, now, annihilated by an order-six differential operator with the unique factorization $L_6 = M_2 \cdot M_4$, where the order-two differential operator $M_2$ has a differential Galois group (included) in $Sp(2, \mathbb{C})$, and where the order-four differential operator $M_4$ has a differential Galois group (included) in $SO(4, \mathbb{C})$. While the (left) differential operator $M_2$ is homomorphic to the order-two differential operator annihilating the diagonal of $1/(1 - (x + y + z))$, we have no interpretation for the ‘dressing’ right factor $M_4$ with respect to the right factor of $\tilde{Q}$.

This example shows that changing the coefficients in front of the monomials to a value that makes the polynomial $Q$ factorizable leads to a reducible differential operator. A more subtle situation is addressed in appendix D.

<footnote>$^{24}$ In particular, in the case where both operators have the same order and the same singularities, the ‘dressing’ operator $N$ in (44) is not homomorphic to the differential operator annihilating the diagonal of $1/Q_2$.</footnote>

<footnote>$^{25}$ For $Q_1$ polynomial of three variables, the ‘dressing’ operator $N$ seems to always have algebraic solutions.</footnote>
**Remark 6.1.** The formal solution of $L_6$ (and of $M_2$) with the highest log-power at the origin is in $\ln(t)^1$, in agreement with the fact that $\tilde{Q}$ depends on $3 = 1 + 2$ variables. The ‘dressing’ differential operator $M_4$ has no logarithmic formal solution around all the singularities, and all these formal solutions are globally bounded. According to Christol’s conjecture [4] these formal solutions are diagonal of rational functions and according to our conjecture (3), the rational functions depend on only two variables and therefore, should be algebraic series.

In fact, the calculation of the $p$-curvature of the differential operator $M_4$ gives zero<sup>26</sup> for every prime $p$, in agreement with algebraic series (according to Grothendieck–Katz conjecture [33]).

## 6.4. Second example

Consider, now, the denominator $Q$

$$Q = (1 - (x+y+z+u)) \cdot (1 - xy-zu), \qquad (47)$$

and the diagonal of the rational function $1/Q$:

$$\mathrm{Diag}\left(\frac{1}{Q}\right) = 1 + 30t + 2958t^2 + 428652t^3 + 72819090t^4 + \dots \qquad (48)$$

This diagonal is annihilated by an order-five differential operator that factorizes as $L_5 = L_3 \cdot L_2$, where $L_3$ has a differential Galois group (included) in $SO(3, \mathbb{C})$, and $L_2$ has a differential Galois group (included) in $Sp(2, \mathbb{C})$. The differential operator $L_3$ is homomorphic with the order-three differential operator annihilating the diagonal of $1/(1 - (x+y+z+u))$. Note that the formal solution of $L_5$ with the highest log-power at the origin, is in $\ln(t)^2$, indicating that we should deal with a minimum number of $4 = 2 + 2$ variables, in agreement with (3).

Here again, while the left factor of the differential operator $L_5$, and the differential operator annihilating the diagonal of the first factor of $Q$, are actually related by operator homomorphism, we have no interpretation on the right factor $L_2$ with respect to the right factor of $Q$. The differential operator $L_2$ has a differential Galois group (included) in $Sp(2, \mathbb{C})$, and carries $\ln(t)^1$ as the formal solution with the highest log-power, meaning that it is, per se, given by the diagonal of a rational function with $3 = 1 + 2$ variables:

$$\begin{aligned} \mathrm{sol}(L_2) &= \frac{1}{\sqrt{1-4t}} \cdot {}_2F_1\left(\left[\frac{1}{4}, \frac{3}{4}\right], [1], \frac{64}{9}t\right) \\ &= \frac{1}{\sqrt{1-4t}} \cdot \mathrm{Diag}\left(\frac{1}{1-c_1x-c_2y-c_3/3 \cdot z^{1/2}}\right) \quad \text{where: } c_1c_2c_3^2 = 1. \end{aligned} \qquad (49)$$

**Remark 6.2.** One should note that the second factor in the polynomial (47), in itself, depends on only two variables $xy$ and $zu$. However, in the product of polynomials given in (47), all the four variables contribute to the diagonal. This is not the kind of situation of section 6.2 where the two polynomials should depend separately on the same number of the same variables for which the diagonal is not blind (see section 1.3). The resulting differential operator $L_5 = L_3 \cdot L_2$, annihilating the diagonal of $1/Q$, with the differential operator $L_2$, shows that there

<footnote n="26"><sup>26</sup> We thank Bostan for calculating the first $p$-curvatures, thus showing that these $p$-curvatures are zero for $7 \le p \le 73$.</footnote>
is a product of polynomials $Q_1 Q_2$ where $Q_1$ and $Q_2$ depend respectively on four and three variables and such that the diagonals of $1/Q$ and of $1/Q_1/Q_2$ identify.

In the previous examples, the denominator polynomial $Q$ factorizes into two polynomials $Q = Q_1 Q_2$, where the number of variables occurring in each polynomial is different. Appendix E addresses the case when the product $Q_1 Q_2$ switches from the situation where both polynomials $Q_j$ carry the same number of variables, to the situation where one of them has a smaller number of variables. Appendix F considers a denominator $Q = Q_1 Q_2$, depending on one parameter $b$, and where the polynomials $Q_1$ and $Q_2$ depend, respectively, on 4 and 3 variables. We address in appendix F the situation where, for one particular value of the parameter $b$, the polynomial $Q$ reduces to a polynomial depending on only three variables instead of the four variables we started with.

# 7. Rational versus algebraic functions and powers of rational functions

In this section we consider some examples of diagonals of algebraic functions, square root of rational functions, and powers of rational functions.

## 7.1. A modular form example

Let us consider the order-two differential operator $L_2$ annihilating the hypergeometric function ${}_2F_1([1/12, 5/12], [1], 1728t)$:

$$L_2 = (1 - 1728t) \cdot t \cdot D_t^2 + (1 - 2592t) \cdot D_t - 60. \quad (50)$$

This order-two differential operator $L_2$ is homomorphic to its adjoint provided$^{27}$ one considers a simple square-root algebraic extension:

$$(1 - 1728t)^{1/2} \cdot \text{adjoint}(L_2) = L_2 \cdot (1 - 1728t)^{1/2}. \quad (51)$$

This hypergeometric function corresponds to the diagonal

$${}_2F_1\left(\left[\frac{1}{12}, \frac{5}{12}\right], [1], 1728t\right) = \text{Diag}(R), \quad (52)$$

of the algebraic function $R$ (depending on two variables with $t = xy$) given by:

$$R = \left( \sqrt{1 - 1728xy} - \sqrt{432} \cdot (x - y) \right)^{-1/6}. \quad (53)$$

According to conjecture (3), we would like to be able to write (52), not as a diagonal of an algebraic function of two variables, but as a rational function of three variables.

The procedure in [34] by Denef and Lipschitz gives, on this example, a rational function $\tilde{R}$ depending on four variables $(x, y, u, v)$ such that (52), the diagonal of $\tilde{R}$ on the four variables $(x, y, u, v)$, identifies with the diagonal of the algebraic function $R$ on the variables $(x, y)$. In this procedure, the denominator of $\tilde{R}$ comes in a factorized form by construction, and each factor does not contain all the variables. For an algebraic function with $n$ variables

<footnote>$^{27}$ Alternatively one can use the ‘Homomorphisms’ command of DEtools in Maple on the symmetric square of adjoint($L_2$) and $L_2$.</footnote>
$(x_1, x_2, \dots, x_n)$, the Denef and Lipschitz procedure [34] will give, as denominator, the product $Q(x_1, x_2, \dots, x_n, u_1) \cdot Q(x_1, x_2, \dots, x_n, u_2) \cdots Q(x_1, x_2, \dots, x_n, u_n)$.

For the hypergeometric function ${}_2F_1([1/12, 5/12], [1], 1728t)$, the procedure sketched in appendix B, is not applicable. However, one may imagine that, introducing a pullback, the general term may be cast in the appropriate form of *ratio of factorials*. One actually has (with a double expansion, index summation change, and summation of the inner sum):

$$ 
\begin{aligned}
{}_2F_1\left(\left[\frac{1}{12}, \frac{5}{12}\right], [1], 1728t \cdot (1 - 432t)\right) \\
&= \sum_{n=0}^{\infty} \sum_{k=0}^{n} (-4)^k \cdot \binom{k}{n-k} \cdot \frac{(1/12)_k (5/12)_k}{(1)_k k!} \cdot (-432t)^n \\
&= \sum_n \frac{(6n)!}{n! (2n)! (3n)!} \cdot t^n.
\end{aligned} 
\quad (54) $$

The last multinomial form leads to (with $t = xyz$):

$$ {}_2F_1\left(\left[\frac{1}{12}, \frac{5}{12}\right], [1], 1728t \cdot (1 - 432t)\right) = \mathrm{Diag}\left(\frac{1}{1 - x - y^{1/2} - z^{1/3}}\right). \quad (55) $$

The order-two differential operator, annihilating the diagonal (55), is not only *homomorphic* with its adjoint, it is self-adjoint. Its differential Galois group is (included) in $Sp(2, \mathbb{C})$, and the formal solution of the differential operator with the highest log-power carries a $\ln(t)^1$, in agreement with the $3 = 1 + 2$ variables of the diagonal in (55), and conjecture (3).

**Remark 7.1.** From (55) we easily get that

$$ {}_2F_1\left(\left[\frac{1}{12}, \frac{5}{12}\right], [1], 1728t\right) = \mathrm{Diag}\left(\frac{1}{1 - \alpha \cdot x - y^{1/2} - z^{1/3}}\right), \quad (56) $$

$$ \text{where } \alpha = \frac{1 - (1 - 1728t)^{1/2}}{864t}, \quad \text{with } t = xyz. \quad (57) $$

*Up to an algebraic function of the product $t = xyz$, we thus have a representation of (52) as a diagonal of a rational function of three variables.*

**Remark 7.2.** Note that the pullback in (55) is precisely the one that matches (55) with one of a *modular form* of appendix B in [35]:

$$ {}_2F_1\left(\left[\frac{1}{12}, \frac{5}{12}\right], [1], 1728 \cdot t \cdot (1 - 432t)\right) = {}_2F_1\left(\left[\frac{1}{6}, \frac{5}{6}\right], [1], 432t\right). \quad (58) $$

## 7.2. From the LGF of 3D s.c to Calabi-Yau number 69

Recall the rational function $1/Q$, depending on four variables, and corresponding$^{28}$ to the LGF of the three-dimensional s.c appendix (A.1.1),

$$ Q = 1 - (x + y + z + u \cdot (xy + xz + yz)), \quad (59) $$

<footnote>$^{28}$ This Heun function (60) can be written as pullbacked ${}_2F_1$ hypergeometric function with algebraic pullbacks (see equations (12), (13) and (14) in [35]).</footnote>
$$ \mathrm{Diag}\left(\frac{1}{Q}\right) = \mathrm{HeunG}\left(\frac{1}{9}, \frac{1}{12}, \frac{1}{4}, \frac{3}{4}, 1, \frac{1}{2}, 4 \cdot t\right)^2. \qquad (60) $$

Considering, now, the diagonal of the reciprocal of $Q^{1/2}$,

$$ \mathrm{Diag}\left(\frac{1}{Q^{1/2}}\right) = 1 + \frac{9}{4}t + \frac{1575}{64}t^2 + \frac{107415}{256}t^3 + \dots \qquad (61) $$

one obtains an annihilating irreducible order-four differential operator, with a differential Galois group included in the symplectic group $Sp(4, \mathbb{C})$, and with $\ln(t)^3$ highest log-power formal solution. According to conjecture (3), a rational function $1/Q_{eq}$, depending on 5 = 3 + 2 variables should exist. It actually reads:

$$ Q_{eq} = 1 - \left( x + y + z + u \cdot (xy + xz + yz) + \frac{v^{1/2}}{4} \right). \qquad (62) $$

The diagonal of $1/Q_{eq}$ identifies with the diagonal of $1/Q^{1/2}$. With the rescale $v^{1/2}/4 \to v^{1/2}$ (i.e. $t \to 16t$) it corresponds to the Calabi-Yau series (number 69 in [10]):

$$ CY_{69} = \sum_{n=0}^{\infty} \frac{(4n)!}{n!^2 (2n)!} \cdot \sum_{k}^{n} \left(\binom{n}{k}\right)^2 \left(\binom{2k}{k}\right) \cdot t^n. \qquad (63) $$

### 7.3. From the LGF of 3D f.c.c to Calabi-Yau number 14

The diagonal of the rational function $1/Q$ with denominator

$$ Q = 1 - (x + y + z + u \cdot (y + z)), \qquad (64) $$

reads:

$$ \mathrm{Diag}\left(\frac{1}{Q}\right) = {}_3F_2\left(\left[\frac{1}{2}, \frac{1}{3}, \frac{2}{3}\right], [1, 1], 108t\right). \qquad (65) $$

This hypergeometric series (65) actually occurs<sup>29</sup> for the LGF of the three-dimensional f.c.c lattice. Let us, now, consider a rational number $\alpha$, and let us introduce the algebraic function $1/Q(\alpha)$:

$$ Q(\alpha) = (1 - (x + y + z + u \cdot (y + z)))^{\alpha}. \qquad (66) $$

One actually obtains for the diagonal of $1/Q(\alpha)$:

$$ \mathrm{Diag}\left(\frac{1}{Q(\alpha)}\right) = {}_4F_3\left(\left[\frac{1}{2}, \frac{\alpha}{3}, \frac{\alpha}{3} + \frac{1}{3}, \frac{\alpha}{3} + \frac{2}{3}\right], [1, 1, 1], 108t\right). \qquad (67) $$

For $\alpha = 1/2$, the diagonal given in (67)

$$ \mathrm{Diag}\left(\frac{1}{Q(1/2)}\right) = 1 + \frac{15}{4}t + \frac{31185}{256}t^2 + \frac{6381375}{1024}t^3 + \dots \qquad (68) $$

<footnote n="29"><sup>29</sup> The solution of the LGF of 3D f.c.c is given in (19), a pullbacked form of (65).</footnote>
is annihilated by an irreducible order-four differential operator, homomorphic to its adjoint, with $\ln(t)^3$ as the maximum power in its formal solutions around the origin. This globally bounded power series (68) is actually the diagonal of a rational function $1/\tilde{Q}(1/2)$ with $5 = 3 + 2$ variables where the polynomial denominator reads:

$$ \tilde{Q}(1/2) = 1 - \left(x + y + z + u \cdot (y + z) + \frac{v^{1/3}}{4}\right). \qquad (69) $$

The diagonal of $1/\tilde{Q}(1/2)$ identifies with the series given in (68). The polynomial (69) (with the change $v^{1/3}/4 \to v^{1/3}$) is the same polynomial as the one given in table 1, which corresponds to Calabi–Yau equation number 14.

**Remark 7.3.** There is a known relation [6, 36, 37] between the LGF of the d-dimensional diamond and the LGF of the (d + 1)-dimensional s.c lattices. The general term of the series, corresponding to the d-dimensional diamond lattice is the square of the multinomial coefficient on d + 1 indices. The general term of the series corresponding to the d-dimensional s.c lattice, is the square of the multinomial coefficient on d indices times the binomial $\binom{2n}{n}$. The origin of this binomial is considered by Guttmann in [6]. The (d + 1)-dimensional hyper-s.c lattice generating function reads (see (37) and (40) in [6], $u = z/(d+1)$):

$$ \sum_{n=0}^{\infty} \frac{1}{4^n} \binom{2n}{n} \cdot \sum_{k_1 + \cdots + k_{d+1} = n} \left( \frac{n!}{k_1! k_2! \cdots k_{d+1}!} \right)^2 \cdot u^{2n}. \qquad (70) $$

The d-dimensional hyper-diamond lattice generating function is actually expressed in terms of sum of squares of (d + 1)-dimensional multinomials, (see (42) and (43) in [6], $u = z/(d+1)$):

$$ \sum_{n=0}^{\infty} \sum_{k_1 + \cdots + k_{d+1} = n} \left( \frac{n!}{k_1! k_2! \cdots k_{d+1}!} \right)^2 \cdot u^{2n}. \qquad (71) $$

Using the relation

$$ \frac{(1/2)_n}{n!} = \frac{1}{4^n} \frac{(2n)!}{n!^2} = \frac{1}{4^n} \binom{2n}{n}, \qquad (72) $$

the two relations (70) and (71) yield, in terms of diagonals of rational functions (of a variable which reads $t = u^2$)

$$ \mathrm{Diag} \left( \frac{1}{Q(d+1,sc)} \right) = \mathrm{Diag} \left( \frac{1}{\sqrt{Q(d,dia)}} \right), \qquad (73) $$

where $Q(d+1,sc)$ is the (denominator) polynomial corresponding to the (d + 1)-dimensional hyper-s.c lattice and $Q(d,dia)$ the (denominator) polynomial corresponding to the d-dimensional hyper-diamond lattice. There is one less variable in $Q(d,dia)$ than in $Q(d+1,sc)$, in the equality (73) between the diagonals of a rational function and the square root of another rational function. This is similar to the relation between the square root of the rational function of the LGF of s.c (resp. f.c.c) and the rational function corresponding to Calabi–Yau number 69 (resp. Calabi–Yau number 14) given above. Other examples are given in appendix G.
### 7.4. Power of rational functions and homomorphisms

Consider the polynomial $Q$ dependent on the parameter $\mu$:

$$Q = 1 - (x + y + z^2 + \mu \cdot xz). \qquad (74)$$

The diagonal of $1/Q$ is annihilated by an irreducible order-four differential operator $L_4$, and the diagonal of the square $1/Q^2$ is annihilated by an irreducible order-four differential operator $N_4$. These two differential operators are actually *homomorphic* [16], giving

$$\mathrm{Diag} \left( \frac{1}{Q^2} \right) = W_3(\mu) \cdot \mathrm{Diag} \left( \frac{1}{Q} \right), \qquad (75)$$

where $W_3(\mu)$ is an intertwiner of order three. For $\mu = 0$, the order-three differential operator $W_3$ reduces to an order-one intertwiner:

$$W_3(\mu = 0) = \frac{5}{2} \cdot t \cdot D_t + 1. \qquad (76)$$

Let us take $\mu = 0$, and generalize the polynomial $Q$ to (with $r$ a positive integer):

$$\tilde{Q} = 1 - (x + y + z^r). \qquad (77)$$

The diagonal of $1/\tilde{Q}$ is annihilated by an order $2r$ differential operator $L_{2r}^{(1)}$. The differential operator, annihilating the diagonal of $1/\tilde{Q}^n$, is also of order $2r$. Let us call it $L_{2r}^{(n)}$. The differential operator $L_{2r}^{(n)}$ is homomorph with the differential operator $L_{2r}^{(n+1)}$, giving:

$$\mathrm{Diag} \left( \frac{1}{\tilde{Q}^{n+1}} \right) = \left( \frac{2r+1}{rn} \cdot t D_t + 1 \right) \cdot \mathrm{Diag} \left( \frac{1}{\tilde{Q}^n} \right), \qquad (78)$$

As far as conjecture (3) is concerned, the rational functions $1/\tilde{Q}$ and $1/\tilde{Q}^n$, have the same minimum number of variables, therefore the formal solutions of the corresponding differential operators *have the same maximum log-power*.

### 7.5. From reciprocal of square root to rational functions

Let us recall<fnref n="30" /> the diagonals $\mathrm{Diag}(1/Q^{1/2})$ given in (61) and (68), where $Q$ depends on four variables, and which identify with the diagonals of $\mathrm{Diag}(1/Q_{\mathrm{eq}})$ given in (62) and (69)

$$\mathrm{Diag} \left( \frac{1}{Q^{1/2}} \right) = \mathrm{Diag} \left( \frac{1}{Q_{\mathrm{eq}}} \right), \qquad (79)$$

$$\text{where} \qquad Q_{\mathrm{eq}} = Q - \frac{1}{4} u^{\alpha}, \qquad (80)$$

with an additional fifth variable $u$, and where $\alpha$ is a rational (not integer) number. Let us show how this occurs with the procedure of appendix B, and show whether the form (80) is general.

<footnote n="30">³⁰ See also examples in appendix G.</footnote>
Consider the multivariate polynomial $Q = 1 - (T_1 + T_2 + \dots + T_n)$, where the $T_j$'s are monomials. The expansion of $1/Q^{1/2}$ reads:

$$ \frac{1}{Q^{1/2}} = \sum \frac{1}{4^{k_0}} \frac{(2k_0)!}{k_0!^2} \cdot (T_1 + T_2 + \dots + T_n)^{k_0} \\ = \sum \frac{1}{4^{k_0}} \frac{(2k_0)!}{k_0!^2} \cdot \frac{(k_1 + k_2 + \dots + k_n)!}{k_1! k_2! \dots k_n!} \cdot T_1^{k_1} T_2^{k_2} \dots T_n^{k_n}. \quad (81) $$

With the constraint $k_0 = k_1 + k_2 + \dots + k_n$, there will be one more factorial (i.e. $k_0!$) in the denominator with respect to the expansion of $1/Q$ given in (8). One has then one additional monomial in the polynomial $Q_{eq}$ if we consider the right hand side of (81) as an expansion of $1/Q_{eq}$.

By performing the diagonal on (81), one factorial in the denominator $k_j!$ takes a value like $k_j! = (rp + \dots)!$, where $p$ is the running index of the diagonal and $r$ an integer. The additional monomial in $Q_{eq}$ will be $v^{1/r}$. Now depending on the monomials $T_j$, there is no reason why the additional variable will not occur elsewhere.

Hereafter, let us display some examples of polynomials $Q$ and $Q_{eq}$ (with $\mu$ a parameter) where the relation (79) holds but with an expression for $Q_{eq}$, different from the simple expression given in (80):

$$ Q = 1 - (x + y + z + \mu x^5 y), \quad (82) $$

$$ Q_{eq} = 1 - \left( x + y + z + \mu x^5 y u^{5/3} + \frac{1}{4} u^{1/3} \right), \quad (83) $$

$$ Q = 1 - (x + y + z + xy + yz + \mu x^3 yz), \quad (84) $$

$$ Q_{eq} = 1 - \left( x + y + z + (xy + yz) \cdot u^{1/3} + \mu x^3 yz u^{4/3} + \frac{1}{4} u^{1/3} \right). \quad (85) $$

Such relations, and relation (80), single out square roots, and cannot be simply generalized to any $N$th roots of polynomials. This special role played by reciprocal of square roots comes, in fact, from the emergence of ratio of factorials of multiple argument in the relation (72).

In view of the expressions of the polynomials $Q_{eq}$ a natural question arises for the case where the polynomial $Q$ is factorizable (in two factors, for instance) $Q = Q_1 Q_2$. Does the additional variable $u$ occurs in one factor, or in both, or as an additive monomial? In the last situation, the equivalent polynomial $Q_{eq}$ may come out non factorizable, and the equivalent differential operator will be irreducible.

Let us consider the polynomial $\tilde{Q} = (1 - x - y - z) \cdot (1 - y - z^2)$ given in (46). Here the variables in the second factor of $\tilde{Q}$ are a subset of the variables in the first factor. The diagonal of $1/\tilde{Q}$ is annihilated by an order-six differential operator with the unique factorization $L_6 = M_2 \cdot M_4$.

The first terms of the diagonal of $1/\tilde{Q}^{1/2}$ are $(t = xyz)$:

$$ \mathrm{Diag} \left( \frac{1}{\tilde{Q}^{1/2}} \right) = 1 + \frac{9}{4} t + \frac{1695}{64} t^2 + \frac{26215}{64} t^3 + \frac{120986775}{16384} t^4 + \dots \quad (86) $$

This globally bounded power series is annihilated by an order-eight differential operator $L_8$. The differential operator $L_8$ is irreducible with a differential Galois group included in $SO(8, \mathbb{C})$. Among the formal solutions, at the origin, of the differential operator $L_8$, two series behave as $\ln(t)^2$, (i.e. the maximum log power). One of these solutions is the series given
in (86) which, according to our conjectures, should be diagonal of a rational function with four variables ($N_v = 2 + 2$).

In order to find such a rational function $1/Q_{eq}$, we follow the procedure in appendix B with the general term of the diagonal of $1/\tilde{Q}^{1/2}$, to obtain:

$$Q_{eq} = 1 - x - y - z + \frac{1}{4} \cdot (2xy + y^2 + 2yz - 4z^2) \cdot u^{1/3} \\ + \frac{1}{2} z^2 \cdot (2x + y + 2z) \cdot u^{2/3} - \frac{1}{4} u^{1/3}. \qquad (87)$$

Consider, now, the polynomial $Q = (1 - x - y - z) \cdot (1 - x - y - z^2)$ where both factors in $Q$ carry the same number of variables. The diagonal of $1/Q$ is annihilated by an order-seven differential operator with the direct sum factorization $L_7 = (L_2 \oplus L_4) \cdot N_1$. The diagonal of $1/Q^{1/2}$

$$\mathrm{Diag} \left( \frac{1}{Q^{1/2}} \right) = 1 + 3t + \frac{195}{4} t^2 + 665 t^3 + \frac{820575}{64} t^4 + \dots \qquad (88)$$

is annihilated by an irreducible order-eight differential operator with differential Galois group included in $SO(8, \mathbb{C})$. One of the formal solutions, at the origin, of the differential operator, behaves as $\ln(t)^2$, (i.e. the maximum log power). The power series in front of $\ln(t)^2$ identifies with the series given in (88), and should be the diagonal of a rational function $Q_{eq}$ with four variables. The polynomial $Q_{eq}$ reads:

$$Q_{eq} = 1 - (x+y) \cdot (z-1) \cdot (z+2) + (x+y)^2 \cdot u^{1/2} \\ + z \cdot (z^2 - z - 1) \cdot u^{-1/2} - \frac{1}{4} \cdot u^{1/2}. \qquad (89)$$

## 8. On the homomorphism to the adjoint assumption

All the examples, displayed in this paper (and many others not given here), confirm the conjecture (3). One assumption for this conjecture to hold, is that the (minimal order) differential operator is *homomorphic to its adjoint*, thus yielding symplectic, or orthogonal, differential Galois groups [19, 38]. This assumption may look as an innocent caveat since, as we underlined in several papers [38], the (minimal order) differential operators annihilating diagonals of rational functions are, almost systematically, homomorphic to their adjoint$^{31}$.

However, some examples of diagonals, whose corresponding (minimal order) differential operators are *not* homomorphic to their adjoint, have been seen to correspond to ${}_3F_2$ candidates to be counterexamples to Christol's conjecture [4, 15]. Such a candidate, for instance, reads [4] (see also [39]):

$${}_3F_2\left(\left[\frac{2}{9}, \frac{5}{9}, \frac{8}{9}\right], \left[1, \frac{2}{3}\right], 27t\right) = \mathrm{Diag}\left(\frac{(1-x-y)^{1/3}}{1-x-y-z}\right), \qquad (90)$$

or

$${}_3F_2\left(\left[\frac{4}{9}, \frac{7}{9}, \frac{10}{9}\right], \left[1, \frac{4}{3}\right], 27t\right) = \mathrm{Diag}\left(\frac{1}{(1-x-y-z) \cdot (1-x-y)^{1/3}}\right). \qquad (91)$$

<footnote>$^{31}$ For reducible differential operator, each factor is homomorphic to its adjoint.</footnote>
Note that the second hypergeometric example (91) is of the form $1/Q^{1/3}$, but not of the form $1/Q^{1/2}$ for which we could expect a representation as a diagonal of a rational function with the introduction of an additional variable (see previous section (7.5)). These two hypergeometric functions are diagonals of quite simple *algebraic functions* of three variables. The order-three differential operators annihilating (90), or (91), are *not* homomorphic to their adjoint. Their differential Galois group is $SL(3, \mathbb{C})$. One of the formal solutions (at the origin) of these two order-three differential operators has the highest logarithmic power $\ln(t)^1$.

Note that a representation of this hypergeometric function (90) (resp. (91)) as a diagonal of a *rational* function of more than *three variables*, is possible [4] using Denef–Lipschitz formulation [34], but a representation as a diagonal of a *rational* function of only $1+2=3$ variables does not seem possible.

Thus these two examples, and all the hypergeometric examples<fnref n="32" /> displayed in [4, 39, 40], do not seem to satisfy conjecture (3). All these examples are outside the framework of this paper, for which the assumption *to be homomorphic to its adjoint* is not superfluous, *but necessary*.

## 9. Conclusion

In this paper, we addressed some properties of the diagonal of rational functions. We restricted the analysis to the *rational* functions of the form $1/Q$.

It seems that the minimal number of variables $N_v$ required to represent a globally bounded D-finite series as a diagonal of rational function, is simply related to the highest power $n$ of the logarithmic formal solutions of the (minimal order) differential operator annihilating the diagonals, by $N_v = n + 2$, provided one *assumes* that this differential operator is *homomorphic to its adjoint*.

Furthermore it is observed that the symplectic, or orthogonal, character of the differential Galois group seems to be related to the *parity* of this highest power $n$ of the logarithmic formal solution.

In the situation where the polynomial $Q$ factorizes in two polynomial factors as $Q = Q_1 \cdot Q_2$, the differential operator annihilating the diagonal of $1/Q$ has either a direct sum or a unique factorization, depending on whether both polynomials $Q_i$ carry all (the same) variables, or not. Furthermore, in the case of a unique factorization, the successive factors in the differential operator are included in the symplectic and orthogonal differential Galois groups, in alternance. Even in these factorized cases conjecture (3) remains valid.

All the results, and educated guess statements, of this paper are just conjectures. One would like to have a demonstration of these various conjectures. At first sight, one can imagine that the number of variables $N_v$ is related to some ‘complexity measurement’ of the (minimal order) differential operator annihilating the diagonal. A very naive measure of the complexity is the order of the operator, and the MUM examples, for which we have a simple relation between the highest power $n$ and the order, seem to confirm such a naive view-point. We show, in this paper, that *this is not the case*. The number of variables $N_v$ is *not related with the order* but with highest log-power $n$. Along this line, and as far as a demonstration of the main conjecture (3) is concerned, let us underline that the crucial role played by the highest log exponent corresponds

<footnote n="32">³² The origin of these examples is explained in [39]. Essentially one discards the algebraic hypergeometric functions and the hypergeometric functions which are Hadamard products. Most are seen to be eventually diagonals of algebraic functions [4, 39], but, for a few of them, it is still open if they are diagonals of algebraic functions (like the original candidate of Christol himself [40]: ${}_3F_2([1/9, 4/9, 5/9], [1, 1/3], 27t)$). The question of the homomorphisms to the adjoint symmetry breaking is addressed in section 5.2 of [16].</footnote>
to the concept of monodromy filtration<fnref n="33" />, which can be introduced even if one is not totally sure that the differential operator is minimal order.

## Data availability statement

All data that support the findings of this study are included within the article (and any supplementary files).

## Acknowledgments

We thank A Bostan and J-A Weil, for discussions on ODE's with algebraic solutions. We thank A Bostan for some p-curvature calculations. We thank G Christol, for monodromy filtration discussions. We thank C Koutschan for LGF discussions.

## Appendix A. Linear differential operators corresponding to Lattice Green functions (LGFs)

### A.1. MUM cases

A.1.1. s.c LGFs in d dimensions. The LGF of the d-dimensional s.c lattice is given by the multiple integral of a rational function with numerator 1, and non factorizable denominator depending on d variables, and on the parameter t.

The diagonal $1/Q$, with polynomial $Q$ given by

$$Q = 1 - \sum_{j=1}^{d} x_j - x_{d+1} \cdot \sum_{j=1}^{d} \prod_{i \neq j}^{d} x_i, \qquad (A.1)$$

depending on $N_v = d + 1$ variables, reproduces the s.c lattice of dimension $d$. The diagonal of $1/Q$ is annihilated by an irreducible differential operator of order $d$, having MUM. The maximum power of the logarithmic formal solutions of the differential operators up to $d = 8$, and the corresponding rational functions follow the two conjectures (3) and (4).

A.1.2. The diamond LGF. The LGF of the three-dimensional diamond lattice is given by the diagonal of $1/Q$, where

$$Q = 1 - xyzu \cdot S(x, y, z) \cdot S\left(\frac{1}{x}, \frac{1}{y}, \frac{1}{z}\right), \qquad (A.2)$$

$$\text{with:} \qquad S(x, y, z) = x + \frac{1}{x} + z \cdot \left(y + \frac{1}{y}\right), \qquad (A.3)$$

With the fourth variable $u$, the number of variables is $N_v = 4$. The corresponding order-three differential operator is irreducible, its differential Galois group is included in the orthogonal group $SO(3, \mathbb{C})$, and its formal solutions with the highest log-power behave as $\ln(t)^2$, in agreement with $N_v = 2 + 2 = 4$ variables.

<footnote n="33">³³ See paragraph 4.2 page 40 of [15].</footnote>
<table><caption>**Table A1.** Minimal number of variables, order, maximum exponent of $\ln(t)^n$ of the formal solutions and differential Galois group for the LGF of the fcc lattice of dimension $d = 2, 3, \dots, 12$.</caption><tr><td>d</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td><td>11</td><td>12</td></tr><tr><td>N<sub>v</sub></td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td><td>11</td><td>12</td><td>13</td></tr><tr><td>Order</td><td>2</td><td>3</td><td>4</td><td>6</td><td>8</td><td>11</td><td>14</td><td>18</td><td>22</td><td>27</td><td>32</td></tr><tr><td>n</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td><td>11</td></tr><tr><td>Sp or SO</td><td>Sp</td><td>SO</td><td>Sp</td><td>SO</td><td>Sp</td><td>SO</td><td>Sp</td><td>SO</td><td>Sp</td><td>SO</td><td>Sp</td></tr></table>

The LGF of the four-dimensional diamond lattice is given by the diagonal of $1/Q$, depending on $N_v = 5$ variables. The corresponding order-four differential operator is irreducible. Its differential Galois group is (included) in the symplectic group $Sp(4, \mathbb{C})$. Its formal solution with the highest log-power behaves as $\ln(t)^3$, in agreement with $N_v = 3 + 2 = 5$ variables.

Likewise, the LGF of the five-dimensional diamond lattice is given by the diagonal of $1/Q$, which depends on $N_v = 6$ variables. The corresponding order-five differential operator is irreducible. Its differential Galois group is (included) in the orthogonal group $SO(5, \mathbb{C})$, and with the formal solution with the highest log-power at the origin behaving as $\ln(t)^4$, in agreement with $N_v = 4 + 2 = 6$ variables.

All these examples are in agreement with the two conjectures (3) and (4).

## A.2. Non-MUM examples: f.c.c LGFs in d dimensions

The LGF of the $d$-dimensional f.c.c lattice is given by the multiple integral of a rational function with numerator 1, and non factorizable denominator depending on $d$ variables. For these lattice Green we have not MUM (except for $d = 2, 3, 4$).

The diagonal $1/Q$ where $Q$ is the polynomial

$$Q(x_1, x_2, \dots, x_d, x_{d+1}) = 1 - \left( \prod_{j=1}^{d+1} x_j \right) \cdot \sum_{j>i} \left( x_i + \frac{1}{x_i} \right) \cdot \left( x_j + \frac{1}{x_j} \right), \quad (A.4)$$

depending on $N_v = d + 1$ variables, reproduces the f.c.c LGF of dimension $d$.

From $d = 2$ to $d = 12$, we denote the corresponding differential operators (the subscript being the order), $G_2^{2D_{fcc}}$, $G_3^{3D_{fcc}}$ [5], $G_4^{4D_{fcc}}$ [6], $G_6^{5D_{fcc}}$ [7], $G_8^{6D_{fcc}}$ [8] and $(G_{11}^{7D_{fcc}}, G_{14}^{8D_{fcc}}, G_{18}^{9D_{fcc}}, G_{22}^{10D_{fcc}}, G_{27}^{11D_{fcc}}, G_{32}^{12D_{fcc}})$ [9].

The differential operators, up to $d = 9$, are known to be irreducible [9]. The differential operators, up to $d = 12$, have formal solutions with the highest log-power at the origin behaving as $\ln(t)^n$. All the differential operators (and their corresponding rational functions) (see table A1) are in agreement with the two conjectures (3) and (4).

All the differential operators up to$^{34}$ $d = 11$ have non globally bounded formal solutions around all the singularities $t \neq 0$. There is an exception for $G_3^{3D_{fcc}}$ which has, at the singularity $t = -1/4$, a globally bounded formal solution in front of $\ln(t + 1/4)^2$, i.e. the same maximum as around the singularity $t = 0$.

<footnote>$^{34}$ For $d = 12$, the differential operator $G_{32}^{12D_{fcc}}$ is known only modulo some primes.</footnote>
## Appendix B. The minimum number of variables in the rational function

For general coefficient of the series written as nested sums of binomials, one may use the integral representation of the binomial

$$ \binom{n}{k} = \frac{1}{2\pi i} \int_C \frac{(1+z)^n}{z^k} \frac{dz}{z}, \qquad (B.1) $$

to write down the rational function [1]. The calculations are straightforward [1], and one obtains, this way, the rational function with as many variables as binomials plus one more variable. Furthermore, the denominator polynomial $Q$ will be in a non factorized form, for more than one summation in the general term of the series.

Let us consider, for instance, the general term of the Calabi–Yau series (number 16 in [10])

$$ \begin{aligned} CY_{16} &= \sum_{n=0}^{\infty} \sum_{k=0}^{n} \binom{2n}{n} \cdot \binom{n}{k}^2 \cdot \binom{2k}{k} \cdot \binom{2n-2k}{n-k} \cdot t^n \\ &= 1 + 8t + 168t^2 + 5120t^3 + \dots \end{aligned} \qquad (B.2) $$

Using the integral representation (B.1), each binomial, and the variable $t$, brings one variable. One obtains

$$ CY_{16} = \mathrm{Diag} \left( \frac{1}{Q_1 \cdot Q_2} \right), \qquad (B.3) $$

where:

$$ \begin{aligned} Q_1 &= 1 - z_0 z_2 z_3 z_4 \cdot (1+z_1)^2 \cdot (1+z_2) \cdot (1+z_3) \cdot (1+z_4)^2, \\ Q_2 &= 1 - z_0 z_5 \cdot (1+z_1)^2 \cdot (1+z_2) \cdot (1+z_3) \cdot (1+z_4)^2. \end{aligned} \qquad (B.4) $$

This diagonal of rational function representation of the Calabi–Yau $CY_{16}$ depends on six variables with a *factorizable* denominator. This is a diagonal of rational function representation of $CY_{16}$, but this simple nested sum of binomials *does not provide the minimal number of variables representation* for the diagonal.

Let us now show how the rational function $1/Q$, in table 1, is obtained for Calabi–Yau $CY_{16}$.

The procedure amounts to casting the general term into the *original form* of the *multinomial* theorem, instead of the previous nested sum of binomials form. This means making the numerator with one factorial by introducing more summations. Converting to factorials, and using the formula

$$ \frac{(2n)!}{n!^4} = \sum_{k=0}^{n} \frac{1}{k!^2 (n-k)!^2}, \qquad (B.5) $$

in the general term of $CY_{16}$, one obtains$^{35}$:

$$ CY_{16} = \sum_{n,k,k_1,k_2} \frac{(2n)!}{k_1!^2 k_2!^2 (k-k_1)!^2 (n-k-k_2)!^2} \cdot t^n \qquad (B.6) $$

<footnote>$^{35}$ This is the original form of the general term given in section 8.1 of [28].</footnote>
with $N_s = 4$ summations. The general term carries one factorial in the numerator, and eight factorials in the denominator. There are, then, $N_m = 8$ monomials in the multinomial expansion of the unknown polynomial denominator $Q = 1 - (T_1 + T_2 + \cdots + T_8)$.

These monomials correspond to each factorial as

$$ 
\begin{array}{l}
T_1 \to k_1, \qquad T_2 \to k_1, \qquad T_3 \to k_2, \qquad T_4 \to k_2, \\
T_5 \to k - k_1, \qquad T_6 \to k - k_1, \\
T_7 \to n - k - k_2, \qquad T_8 \to n - k - k_2
\end{array}
\qquad (B.7) $$

and satisfy

$$ \frac{T_1 T_2}{T_5 T_6} = 1, \qquad \frac{T_3 T_4}{T_7 T_8} = 1, \qquad \frac{T_5 T_6}{T_7 T_8} = 1, \qquad T_7 T_8 = t, \qquad (B.8) $$

which give:

$$ T_2 = \frac{t}{T_1}, \qquad T_4 = \frac{t}{T_3}, \qquad T_6 = \frac{t}{T_5}, \qquad T_8 = \frac{t}{T_7}. \qquad (B.9) $$

The variable $t$ is the product of all the variables. There are $N_m - N_s = 4$ unfixed $T_j$. The number of variables should be greater, or equal, to 4. Guided by our conjectures, let us, first, assume there are five variables, i.e. $t = xyzuv$. Choosing $T_1 = x$, $T_3 = y$, $T_5 = z$ and $T_7 = u$, we obtain the polynomial $Q$ given in table 1. On may also choose $T_1 = x$, $T_3 = y$, $T_5 = xu$ and $T_7 = yv$, to obtain

$$ Q = 1 - x \cdot (1 + u + uz + uzv) - y \cdot (1 + v + vz + vzu), \qquad (B.10) $$

and the diagonal of $1/Q$, identifies with $CY_{16}$.

Assume, now, that we start with six variables, i.e. $t = xyzuvw$, and fix $T_1 = x$, $T_3 = y$, $T_5 = z$ and $T_7 = u$. One obtains the polynomial

$$ Q = 1 - (x+y+z+u + vw \cdot (xyz + yzu + zux + uxy)), \qquad (B.11) $$

where it is clear that the product $vw$ stands for just one variable.

**Remark B.1.** If we take the value $N_m - N_s = 4$ as the number of variables, one obtains $Q = 1 - (x + y + z + u + xyz + yzu + zux + uxy)$. The diagonal of $1/Q$ does not identify with $CY_{16}$. The corresponding differential operator is of order four, is irreducible, with a differential Galois group included in $Sp(4, \mathbb{C})$, and the most logarithmic singular formal solution is in $\ln(t)^2$.

**Remark B.2.** Assume that, in the numerator of (B.6), one has $a^{k_1} (2n)! t^n$, the first condition in (B.8) changes to

$$ \frac{T_1 T_2}{T_5 T_6} = 1 \quad \longrightarrow \quad \frac{T_1 T_2}{T_5 T_6} = a \qquad (B.12) $$

and the denominator polynomial $Q$ becomes:

$$ 1 - (x + y + z + u + v \cdot (xzu + xyu + xyz + ayzu)), \qquad (B.13) $$
**Remark B.3.** Let us recall the examples with a physical origin, displayed in this work, namely the LGF examples (see section 3) and the Calabi–Yau examples (see section 5). For these examples, we claim that we found the minimal number of variables $N_v$ necessary to represent the series as diagonal of rational functions.

On the LGF examples our minimal number corresponds to the (physical) dimension of the lattice. We have seen in remark 7.3 that a smaller number of variables amounts, in fact, to moving from diagonal of rational functions to diagonal of square roots of rational functions.

For the Calabi–Yau examples, the general terms $A_n$, displayed in table 1, can be written in the form of a multinomial expansion. The structure of the general term determines the number of monomials (see (7) and (8)) necessary to reproduce the diagonal series. We can perform a multinomial expansion with unknown $T_j$'s in the spirit of the Calabi–Yau example $CY_{16}$ given above. Following (B.7) the monomials correspond to each factorial in the denominator of the multinomial expansion, and one obtains equations, like (B.8) and (B.9), on the unknown $T_j$'s. We tried to solve these Equations (except for $CY_3$, $CY_{17}$, $CY_{18}$, $CY_{19}$, which have too many terms), with less than five variables. If there is a solution of the equations like (B.8) and (B.9), one actually sees that the diagonal of the corresponding rational function does not identify with the original series of the Calabi–Yau example. This rational function, with less variables, is thus discarded.

Our confidence on the fact that we actually have the minimal number of variables $N_v$ in the examples, related or not related to physics, displayed in this paper, relies on such (or similar) calculations.

## Appendix C. The factors occurring in the $\chi^{(n)}$'s of the Ising model

The differential operators corresponding to the $n$-particles contributions to the magnetic susceptibility of the Ising model, namely the $\chi^{(n)}$'s, and especially their decomposition in products and direct sums of many factors are recalled below (the subscript denote the order of the differential operator):

$$
\begin{align*}
\chi^{(1)}, &\quad L_1, \\
\chi^{(2)}, &\quad L_2, \\
\chi^{(3)}, &\quad L_7 = L_1 \oplus (Y_3 \cdot Z_2 \cdot N_1), \\
\chi^{(4)}, &\quad L_{10} = L_2 \oplus (\tilde{L}_4 \cdot L_4^{(4)}), \qquad L_4^{(4)} = L_{1,3} (L_{1,2} \oplus L_{1,1} \oplus D_t), \\
\chi^{(5)}, &\quad L_{33} = L_7 \oplus (L_5 L_{12} \cdot \tilde{L}_1) \cdot (V_2 \oplus (Z_2 \cdot N_1) \oplus (F_3 \cdot F_2 \cdot L_1^s)), \\
\chi^{(6)}, &\quad L_{52} = L_{10} \oplus (L_6 \cdot \tilde{L}_2 \cdot L_{21}) \cdot \left(\left(D_t - \frac{1}{t}\right) \oplus L_3 \oplus L_4^{(4)} \oplus (L_4 \tilde{L}_3 \cdot L_2^e)\right),
\end{align*}
$$

All the factors, occurring in the differential operators, have been shown to be such that their differential Galois group is either symplectic or orthogonal (see [19] and references therein).

The blocks of factors (of order greater than 1) with unique factorization (with their corresponding differential Galois group), are:

$$
\begin{array}{l@{\qquad}l@{\qquad}l}
Y_3 Z_2, & Y_3 \to SO(3, \mathbb{C}), & Z_2 \to Sp(2, \mathbb{C}) \\
L_5 L_{12}, & L_5 \to SO(5, \mathbb{C}), & L_{12} \to Sp(12, \mathbb{C}), \\
F_3 F_2, & F_3 \to SO(3, \mathbb{C}), & F_2 \to Sp(2, \mathbb{C}), \\
L_6 \tilde{L}_2 L_{21}, & L_6 \to SO(6, \mathbb{C}), & \tilde{L}_2 \to Sp(2, \mathbb{C}), \qquad L_{21} \to SO(21, \mathbb{C}), \\
L_4 \tilde{L}_3 L_2^e, & L_4 \to Sp(4, \mathbb{C}), & \tilde{L}_3 \to SO(3, \mathbb{C}), \qquad L_2^e \to Sp(2, \mathbb{C}).
\end{array}
$$
Each time we deal with a block of factors with unique factorization, the differential Galois groups of the differential operators inside the block, are, in alternance, (included in) symplectic, or orthogonal, differential Galois groups.

Let us note that we have shown [1] that the $\chi^{(n)}$'s are actually diagonals of rational functions. However, for each differential operator factor in the blocks, it is quite hard to find the corresponding rational (or algebraic) function whose diagonal is annihilated by this differential operator factor. Conjecture (3), can just be used to determine the minimum number of variables in the rational function. This should, also, be consistent with the fact that only the series in front of $\ln(t)^n$, with $n$ the maximum exponent, are globally bounded.

Consider, for instance, the order-twelve differential operator $L_{12}$ occurring in the factorization of $L_{33}$, the differential operator annihilating $\chi^{(5)}$. Among the formal solutions of $L_{12}$ around the origin, there are two series in front of $\ln(t)^3$, (the exponent $n=3$ being the maximum), which are, both, globally bounded. According to conjecture (3), these two globally bounded series solutions of the differential operator $L_{12}$, should be diagonal of rational functions of $N_v = 3+2=5$ variables. Around the singularity $t=\infty=1/s$, there are also two series in front of $\ln(s)^3$, (the same value of the exponent), and these series are also globally bounded. At the singularity $t=1/4$, there is one series in front of $\ln(t-1/4)^3$ among the formal solutions, and this series is actually globally bounded. Around each of the other singularities (with rational values), i.e. $t = -1/4, -1/2, -1, 1$, the maximum of power log in the formal solutions is, respectively, 2, 1, 1, 1, and all these series are not globally bounded. For the singularities $t=t_s$, roots of polynomials of degree 2, 3 and 4, we only checked that, for each, the formal solutions behave at the most as $\ln(t-t_s)^1$.

## Appendix D. Other examples

Here, we give some examples that contradict, at first sight, the affirmation that the diagonal of $1/Q$ with non factorizable (resp. factorizable) $Q$ over the rationals, is annihilated by an irreducible (resp. factorizable) differential operator.

The three rational functions $1/Q_j$ with $Q_j$ given by

$$Q_1 = 1 + (x + y + z + xy + yz - x^3 yz), \qquad (D.1)$$

$$Q_2 = 1 - (x + y + z + x^4 y), \qquad (D.2)$$

$$Q_3 = 1 - (x + y + z + x^2 \cdot (y + z)), \qquad (D.3)$$

rule out the irreducibility statement. The diagonal of the rational function $1/Q_j$ is annihilated by an order-four differential operator $L_4^{(j)}$ which factorizes as a direct sum of two order-two differential operators ($L_2^{(j)}$ and $M_2^{(j)}$):

$$L_4^{(j)} = L_2^{(j)} \oplus M_2^{(j)}. \qquad (D.4)$$

The three examples follow exactly the same features. For the three examples, one has

$$\mathrm{sol}(L_2^{(j)}) = \mathrm{Diag}\left(\frac{1+x}{Q_j}\right), \quad \mathrm{sol}(M_2^{(j)}) = \mathrm{Diag}\left(\frac{1-x}{Q_j}\right), \qquad (D.5)$$

and for the three examples, their solutions $\mathrm{sol}(L_2^{(j)})$ and $\mathrm{sol}(M_2^{(j)})$ can be written as pullbacked hypergeometric function ${}_2F_1([1/12, 5/12], [1], \bullet)$ with an algebraic prefactor (see [41], for $Q_1$).
Note however, that when the polynomials $Q_j$ are considered with generic values of the coefficients in front of the monomials, the resulting differential operators, annihilating the diagonal of $1/Q_j$, are of order four, and are irreducible.

Introducing a parameter $\mu$ in $Q_1$

$$Q_1(\mu) = 1 + (x+y+z+xy+yz-\mu \cdot x^3yz), \qquad (D.6)$$

and using the method of factorization of differential operators modulo primes (see section 4 in [22] and remark 6 in [19]), one finds that the differential operator annihilating the diagonal Diag(1/Q_1(\mu)) factorizes for only two values of $\mu$. The trivial $\mu = 0$ where $Q_1(0) = (1+x+z)(1+y)$, and the particular value $\mu = 1$. The example with $Q_1(\mu = 1)$ is presented in [41] and analyzed via the notion of split Jacobian [42].

Note that either irreducible or factorizable in direct sum, the differential operators $L_2^{(j)}$, $M_2^{(j)}$ have their differential Galois groups included in symplectic groups, and their formal solution with the highest log-power behaves as $\ln(t)^1$, indicating a minimal number $N_v = 3 = 1 + 2$ of variables occurring in the rational function $1/Q_j$. This is also the case for the order-four irreducible differential operator corresponding to the diagonal of $1/Q_1(\mu)$, for generic values of the parameter $\mu$.

## Appendix E. Direct sum versus unique factorization for the diagonal of $1/Q$, with $Q = Q_1Q_2$

The examples of section 6 dealt with a polynomial $Q$ that factorizes as $Q_1^{(c)}Q_2$, where the polynomial $Q_1^{(c)}$ contains all the variables, and the polynomials $Q_2$ has a smaller number of variables. Call $L_q$, the differential operator annihilating the diagonal of $1/Q_1^{(c)}$. The differential operator, annihilating the diagonal of $1/Q_1^{(c)}/Q_2$, appears with a unique factorization as $N_q \cdot M_n$, where $N_q$ is homomorphic with $L_q$. There is no known relation between the differential operator, corresponding to $1/Q_2$, and the 'dressing' differential operator $M_n$.

### E.1. Direct sum

Let us consider the situation where $Q$ factorizes as $Q_1^{(c)}Q_2^{(c)}$, and where both polynomials $Q_1^{(c)}$ and $Q_2^{(c)}$ carry all the variables. Call $L_q$, $L_n$ and $\mathcal{L}_m$ the differential operators annihilating respectively, Diag(1/Q_1^{(c)}), Diag(1/Q_2^{(c)}) and Diag(1/Q_1^{(c)}/Q_2^{(c)}). Since the polynomials $Q_1^{(c)}$ and $Q_2^{(c)}$ are on an equal footing in terms of number of variables, one may expect the differential operator $\mathcal{L}_m$ to have a left factor homomorphic to $L_q$ as well as another left factor homomorphic to $L_n$. This means that $\mathcal{L}_m$ has a factorization in direct sum between the homomorphic differential operators of $L_q$ and $L_n$.

### E.2. From direct sum to unique factorization

Let us see how the direct sum actually reduces to a simple product, when one of the factors $Q_j$ has less variables. Take the rational functions $1/Q_1$ and $1/Q_2$ with the polynomials:

$$Q_1 = 1 - (x+y+z), \qquad Q_2(\alpha) = 1 - (x+y+\alpha \cdot z+xy). \qquad (E.1)$$
The diagonal Diag(1/Q₁) (resp. Diag(1/Q₂(α))) is annihilated by an order-two differential operator L₂⁽¹⁾ (resp. L₂⁽²⁾). The diagonal of 1/Q₁/Q₂(α)

$$ 
\begin{aligned}
& \operatorname{Diag} \left( \frac{1}{Q_1 Q_2} (\alpha) \right) \\
& = 1 + (13 + 14\alpha) \cdot t + (241 + 273\alpha + 306\alpha^2) \cdot t^2 + \dots \quad (\text{E.2})
\end{aligned}
$$

is annihilated by a (quite large) order-six differential operator L₆, depending on α, and it factorizes in a direct sum as:

$$ L_6 = (N_2^{(1)} \oplus N_2^{(2)}) \cdot H_2 = (N_2^{(1)} \cdot H_2) \oplus (N_2^{(2)} \cdot H_2). \quad (\text{E.3}) $$

The order-two differential ‘dressing’ operator H₂ has two algebraic solutions:

$$ \operatorname{sol}(H_2) = \frac{1}{\sqrt{t \cdot \sqrt{p_2}}} \cdot (\operatorname{const.} A^{1/4} + \operatorname{const.} (t - \alpha + 1) \cdot A^{-1/4}), \quad \text{where:} $$

$$ A = p_1 + 4 \cdot (2\alpha - 3) \cdot \sqrt{\alpha - 1} \cdot \sqrt{t \cdot \sqrt{p_2}}, \quad \text{where:} $$

$$ \begin{aligned} p_1 &= t^2 + 2 \cdot (\alpha - 1) \cdot (4\alpha - 5) \cdot (4\alpha - 7) \cdot t + (\alpha - 1)^2, \\ p_2 &= t^2 + 2 \cdot (\alpha - 1) \cdot (8\alpha^2 - 24\alpha + 17) \cdot t + (\alpha - 1)^2. \end{aligned} \quad (\text{E.4}) $$

The order-two differential operator N₂⁽¹⁾ (resp. N₂⁽²⁾) is homomorphic with the differential operator L₂⁽¹⁾ (resp. L₂⁽²⁾). The differential operator L₂⁽¹⁾ does not depend on α, but the inter-twiner W₁⁽¹⁾ does:

$$ N_2^{(1)}(\alpha) \cdot W_1^{(1)}(\alpha) = \tilde{W}_1^{(1)}(\alpha) \cdot L_2^{(1)}. \quad (\text{E.5}) $$

From the direct-sum decomposition (E.3), the diagonal of 1/Q₁/Q₂(α) which is annihilated by the differential operator L₆, reads:

$$ \operatorname{Diag} \left( \frac{1}{Q_1 Q_2 (\alpha)} \right) = \operatorname{sol} (N_2^{(1)} (\alpha) \cdot H_2 (\alpha)) + \operatorname{sol} (N_2^{(2)} (\alpha) \cdot H_2 (\alpha)). \quad (\text{E.6}) $$

For α = 0 the analytical solution (at t = 0) of N₂⁽¹⁾(α) · H₂(α) is the analytical solution (at t = 0) of H₂(α) for α = 0, namely the algebraic series

$$ \mathcal{A} = 1 + 21t + 561t^2 + 16213t^3 + 487521t^4 + 15015573t^5 + \dots \quad (\text{E.7}) $$

together with the constant function. The other solution of H₂(α), for α = 0, is not analytic at t = 0:

$$ S_2 = t^{-1/2} \cdot (1 + 13t + 321t^2 + 8989t^3 + \dots). \quad (\text{E.8}) $$

The analytical solutions (at t = 0) of N₂⁽¹⁾(α) · H₂(α) for α = 0 are, the analytical solution (at t = 0) of H₂(α) for α = 0, namely (E.7), together with the series:

$$ S = t + 40t^2 + 1400x^3 + 47110t^4 + 1560328t^5 + \dots \quad (\text{E.9}) $$
The diagonal series (E.2) for $\alpha = 0$ reads:

$$1 + 13t + 241t^2 + 5013t^3 + 110641t^4 + 2532949t^5 + \dots \qquad (E.10)$$

which is nothing but the following combination of (E.7) and (E.9):

$$\mathrm{Diag} \left( \frac{1}{Q_1 \cdot Q_2 (\alpha) = 0} \right) = \mathcal{A} - 8 \cdot \mathcal{S}. \qquad (E.11)$$

This series is, thus, annihilated by only the differential operator $N_2^{(1)} \cdot H_2$, for $\alpha = 0$, with no need of the differential operator $N_2^{(2)} \cdot H_2$ for $\alpha = 0$.

**Remark E.1.** Note that the ‘spurious’ differential operator $N_2^{(2)} \cdot H_2$, actually factorizes as a direct sum for $\alpha = 0$

$$N_2^{(2)} \cdot H_2 (\alpha = 0) = D_t \oplus (L_1 \cdot H_2 (\alpha = 0)), \qquad (E.12)$$

where the order-one differential operator $L_1$ annihilates a rational solution.

## Appendix F. Rational function from four variables to three variables as a parameter varies

The example in section 4.3 shows that, depending on the number (four or three) of variables on which the diagonal is performed, we obtain, in the formal solutions of the corresponding differential operators, the power 2 (i.e. $\ln(t)^2$) or the power 1 (i.e. $\ln(s)^1$) in agreement with conjecture (3).

Here, we consider an example (depending on one parameter) where the diagonal is performed on all the four variables, and the result, for some value of the parameter ($b = 2$ in appendix F.2 below), will indicate that we deal, in fact, with only three variables in agreement with the power $\ln(t)^1$ obtained in the formal solutions.

Let us consider the diagonal of the following rational function of four variables $x, y, z, u$, and a parameter $b$

$$\mathrm{Diag} \left( \frac{1}{Q_1 Q_2} \right) = 1 + 20 \cdot (b+2) \cdot t + 756 \cdot (b^2 + 3b + 6) \cdot t^2 + 34320 \cdot (b^3 + 4b^2 + 10b + 20) \cdot t^3 + \dots \qquad (F.1)$$

where:

$$Q_1 = 1 - x - y - z - u, \qquad Q_2 = 1 - x - y - b \cdot z. \qquad (F.2)$$

The power series (F.1) is annihilated (for generic values of $b$) by an order-six differential operator with the unique factorization

$$L_6(b) = L_3(b) \cdot L_2(b) \cdot L_1 \qquad (F.3)$$

where the order-one differential operator, $L_1$, reads $D_t + 1/2/t$.

The order-three differential operator $L_3(b)$ has the following hypergeometric solution:

$$\frac{1}{t^2 \cdot (1 - b + 64b^2 \cdot t)} \cdot {}_3F_2 \left( \left[ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \right], [1, 2], 256 \cdot t \right). \qquad (F.4)$$
The formal series solutions of $L_3(b)$ have, at most, a $\ln(t)^2$.

The order-two differential operator $L_2(b)$ has the following $_2F_1$ hypergeometric solution:

$$ \frac{1}{t} \cdot {}_2F_1\left(\left[\frac{3}{4}, \frac{5}{4}\right], [1], \frac{64 \cdot b^2}{b-1} \cdot t\right). \qquad (F.5) $$

The formal series solutions of $L_2(b)$ have, at most, a $\ln(t)^1$ logarithmic power.

The formal series solutions of $L_6(b)$ have, at most, a $\ln(t)^2$ logarithmic power. This is in agreement with conjecture (3), and the fact that (F.1) corresponds to a rational function of $2+2=4$ variables.

Let us introduce the order-three differential operator $U_3$, annihilating

$$ \mathrm{Diag}\left(\frac{1}{Q_1}\right) = {}_3F_2\left(\left[\frac{1}{4}, \frac{2}{4}, \frac{3}{4}\right], [1, 1], 256t\right). \qquad (F.6) $$

This order-three differential operator reads:

$$ U_3 = (1 - 256t) \cdot t^2 \cdot D_t^3 + 3 \cdot (1 - 384t) \cdot t \cdot D_t^2 \\ + (1 - 816t) \cdot D_t - 24. \qquad (F.7) $$

The formal series solutions of $U_3$ have, at most, a $\ln(t)^2$ logarithmic power, in agreement with conjecture (3) and the fact that $1/Q_1$ is a rational function of four variables. The order-three differential operator $L_3(b)$, in the factorization given in (F.3), is actually homomorphic to the order-three differential operator $U_3$.

All the results above are for the generic values of the parameter $b$. In the sequel we consider the situation with the values of $b$ for which the series (F.1) will be annihilated by a differential operator of order less than six. These values of $b$ are$^{36}$ $b = 0, b = 1$ and $b = 2$.

### F.1. The $b = 0$ and $b = 1$ cases

For the value of the parameter $b = 0$, the factorization of the order-six differential operator $L_6(b)$ becomes

$$ L_6(b=0) = \left( \left( D_t + \frac{1}{t} \right) \cdot D_t \right) \oplus (N_3 \cdot L_1). \qquad (F.8) $$

For $b = 0$ the series (F.1) is annihilated by the order-four differential operator $N_3 \cdot L_1$, whose formal solutions, at the origin carry the maximum exponent $\ln(t)^2$, indicating, according to conjecture (3), that we deal with a rational function with $4 = 2 + 2$ variables.

For the value of the parameter $b = 1$, the factorization of the order-six differential operator $L_6(b)$ reads:

$$ L_6(b=1) = \left( D_t + \frac{1}{2t} \right) \oplus \left( D_t + \frac{3}{4t} \right) \oplus \left( D_t + \frac{5}{4t} \right) \oplus M_3. \qquad (F.9) $$

For $b = 1$ the series (F.1) is annihilated by the order-three differential operator $M_3$, whose formal solutions at the origin carry the maximum exponent $\ln(t)^2$ indicating, according to conjecture (3), that we deal with a rational function with $2+2=4$ variables.

<footnote>$^{36}$ These values of $b$ can be obtained using the method of factorization of differential operators modulo primes (see section 4 in [22] and remark 6 in [19]). They can also be obtained by reducing the four singularities $t = \infty, 0, 1/256, (b-1)/64/b^2$ of $L_6(b)$ to the three singularities $t = \infty, 0, 1/256$.</footnote>
## F.2. The b = 2 case

For $b = 2$ the factorization of the order-six differential operator $L_6(b)$ becomes:

$$L_6(b=2) = V_3 \oplus (L_2(b=2) \cdot L_1). \qquad (F.10)$$

The series (F.1), for the parameter $b = 2$, is annihilated by the order-three differential operator $L_2(b=2) \cdot L_1$ with the hypergeometric solution:

$$\mathrm{sol}(L_2(b=2) \cdot L_1) =_3 F_2 \left( \left[ \frac{1}{2}, \frac{3}{4}, \frac{5}{4} \right], \left[ 1, \frac{3}{2} \right], 256t \right) \qquad (F.11)$$

The formal solutions at the origin of $L_2(b=2) \cdot L_1$ carry the maximum exponent $\ln(t)^1$ indicating, according to conjecture (3), that we deal with a rational function with $3 = 1+2$ variables, while the rational function we started with, was dependent on four variables. The power series (F.11) should, then, be the diagonal of a rational function depending on only three variables.

Introducing the polynomial $Q_3 = 1 - x - y - b \cdot u$, the partial fraction form of $1/Q_1/Q_2$ in the variable $z$, reads for $b = 2$:

$$\frac{1}{Q_1 Q_2} = - \frac{1}{Q_1 Q_3} + \frac{2}{Q_2 Q_3}. \qquad (F.12)$$

The diagonal of $1/Q_1/Q_2$, and the diagonal of $1/Q_1/Q_3$, are identical, by the symmetry $(z, u) \to (u, z)$. We end up with

$$\begin{aligned} \mathrm{Diag} \left( \frac{1}{Q_1 Q_2} \right) &= \mathrm{Diag} \left( \frac{1}{Q_2 Q_3} \right) \\ &= \mathrm{Diag} \left( \frac{1}{(1-x-y-2z) \cdot (1-x-y-2u)} \right). \end{aligned} \qquad (F.13)$$

It remains to show that this diagonal depends, in fact, on only three variables instead of four variables:

$$\begin{aligned} \mathrm{Diag} \left( \frac{1}{Q_2 Q_3} \right) &= \mathrm{Diag} \left( \frac{1}{(1-x-y)^2} \cdot \sum_{i,j} \frac{(2z)^i (2u)^j}{(1-x-y)^{i+j}} \right) \\ &= \mathrm{Diag} \left( \frac{1}{(1-x-y)^2} \cdot \sum_{k=0}^{\infty} \frac{(4zu)^k}{(1-x-y)^{2k}} \right). \end{aligned} \qquad (F.14)$$

The sum on the index $k$ gives:

$$\mathrm{Diag} \left( \frac{1}{Q_2 Q_3} \right) = \mathrm{Diag} \left( \frac{1}{(1-x-y)^2 - 4zu} \right). \qquad (F.15)$$

As far as the diagonal is concerned, the product $zu$ stands for only one variable. The diagonal given in (F.1) for $b = 2$, which is the series given in (F.11) is the diagonal of a rational function depending on three variables, as the power of the logarithm in the formal solutions of $L_2(b = 2) \cdot L_1$ indicates.
**Remark F.1.** In the factorization of $L_6(b)$ for $b = 2$, given in (F.10), the differential operator $L_2(b = 2) \cdot L_1$ annihilates the series (F.1) for the value $b = 2$. Therefore, the differential operator $V_3$ becomes ‘spurious’. The analytical solution (at the origin) of the differential operator $V_3$ reads

$$ \mathrm{sol}(V_3) = {}_3F_2\left(\left[\frac{2}{4}, \frac{3}{4}, \frac{5}{4}\right], [1, 1], 256t\right), \qquad (\mathrm{F.16}) $$

to be compared with the solution (F.6) of the differential operator $U_3$. The differential operators $U_3$ and $V_3$ are actually *homomorphic*

$$ (1 - 256t) \cdot t^2 \cdot V_3 \cdot \left(t \cdot D_t + \frac{1}{4}\right) = \left(t \cdot D_t + \frac{5}{4}\right) \cdot (1 - 256t) \cdot t^2 \cdot U_3, \quad (\mathrm{F.17}) $$

which shows (see section 7.4 ) that (F.16) is in fact:

$$ \mathrm{sol}(V_3) = \mathrm{Diag}\left(\frac{1}{Q_1^2}\right). \qquad (\mathrm{F.18}) $$

At the value of $b = 2$, even if the differential operator $V_3$ is spurious with respect to the series (F.1) for $b = 2$, the differential operator $L_6(b)$ has kept, for $b = 2$, some ‘memory’ of the rational function $1/Q_1$, through the spurious operator $V_3$.

## Appendix G. Square root of rational functions versus Calabi–Yau equations

As in sections 7.2 and 7.3, we give, here, two more examples where the diagonal of square root of a rational function produces a Calabi–Yau equation.

### G.1. The square root of the rational function of the LGF 4D simple cubic (s.c)

Recall the rational function $1/Q$, depending on 5 variables, and corresponding to the LGF of the 4-dimensional s.c lattice:

$$ \begin{gathered} Q = 1 - (x+y+z+u+v \cdot (xyz+xyu+xzu+yzu)), \\ \mathrm{Diag}\left(\frac{1}{Q}\right) = 1 + 8t + 168t^2 + 5120t^3 + 190120t^4 + \dots \end{gathered} \qquad (\mathrm{G.1}) $$

Considering, now, the diagonal of the *square root* of this rational function (G.1)

$$ \mathrm{Diag}\left(\frac{1}{Q^{1/2}}\right) = 1 + 3t + \frac{735}{16}t^2 + 1155t^3 + \frac{152927775}{4096}t^4 + \dots $$

one obtains an annihilating irreducible order-five differential operator $L_5$, with a differential Galois group included in the orthogonal group $SO(5, \mathbb{C})$, and with $\ln(t)^4$ highest log-power formal solution at the origin. According to conjecture (3), a rational function $1/Q_{eq}$, depending on 6 = 4 + 2 variables should exist. With the introduction of an extra variable $w$, it actually reads:

$$ Q_{eq} = Q - \frac{w^{1/2}}{4}. \qquad (\mathrm{G.2}) $$
The diagonal of the algebraic function $1/Q^{1/2}$ identifies with the diagonal of the rational function $1/Q_{eq}$.

The differential operator $L_5$ has MUM, and is actually the exterior square of an order-four differential operator, $L_5 = \text{ext}^2(L_4)$. The differential operator $L_4$ also has MUM, at $t = 0$, the indicial exponents being four times $1/2$. Let us introduce the differential operator $N_4$:

$$N_4 = L_4 \cdot \sqrt{t} \cdot (1 - 16t)^{1/4} \cdot (1 - 64t)^{1/4}. \qquad (G.3)$$

With the scaling $t \to 16t$, the differential operator $N_4$ reads (with $\theta = tD_t$):

$$\begin{aligned} N_4 = \theta^4 & - 4 \cdot t \cdot (960 \theta^4 + 640 \theta^3 + 574 \theta^2 + 254 \theta + 41) \\ & + 16 \cdot t^2 \cdot (356352 \theta^4 + 451328 \theta^2 + 475136 \theta^3 + 199424 \theta + 34257) \\ & - 2^{12} \cdot t^3 \cdot (1003520 \theta^4 + 2007040 \theta^3 + 2098048 \theta^2 + 1043328 \theta + 198453) \\ & + 2^{22} \cdot t^4 \cdot (356352 \theta^4 + 950272 \theta^3 + 1126912 \theta^2 + 618752 \theta + 124913) \\ & - 2^{37} \cdot t^5 \cdot (2\theta + 1) \cdot (960\theta^3 + 2720\theta^2 + 2854\theta + 987) \\ & + 2^{48} \cdot t^6 \cdot (2\theta + 1) \cdot (2\theta + 3) \cdot (4\theta + 3) \cdot (4\theta + 5). \end{aligned} \qquad (G.4)$$

To be of Calabi-Yau type, the differential operator $N_4$ must satisfy some conditions [31, 43]. The differential Galois group of $N_4$ is included in $Sp(4, \mathbb{C})$, the differential operator $N_4$ has MUM, and at the infinity, the indicial exponents $1/2, 3/4, 5/4, 3/2$ are such that $1/2 + 3/2 = 3/4 + 5/4 = 2$ is a rational. Moreover, the coefficients of the power series given below in (G.5), and the instanton numbers given below in (G.11), should be integers.

The formal solutions, at the origin, of the differential operator $N_4$ are

$$S_0 = 1 + 164t + 66972t^2 + 38050160t^3 + \dots \qquad (G.5)$$

and three formal solutions $S_j$, ($j = 1, 2, 3$) in the form of the set (1) and behaving as $S_j = S_0 \ln(t)^j + \dots$

Let $z = S_1/S_0$, the nome $q$, defined as $q = \exp(z)$, has the expansion

$$q = t + 360t^2 + 188244t^3 + 119619168t^4 + \dots \qquad (G.6)$$

and the mirror map reads:

$$t = q - 360q^2 + 70956q^3 - 14059968q^4 + \dots \qquad (G.7)$$

The Yukawa coupling defined by

$$K(q) = \frac{d^2}{dz^2} \left( \frac{S_2}{S_0} \right), \qquad (G.8)$$

reads:

$$K(q) = 1 - 128q - 41984q^2 - 13919744q^3 - 4141162496q^4 + \dots \qquad (G.9)$$

The Yukawa coupling, expanded in a Lambert series, reads

$$K(q) = 1 + \sum_{j=1}^{\infty} n_j \frac{j^3 q^j}{1 - q^j}, \qquad (G.10)$$
and gives the ‘instantons numbers’ $n_j, j = 1, 2, \dots$

$$ -128, -5232, -\frac{1546624}{3}, -64705008, -7960717440, -1089730087792, \dots \quad (G.11) $$

where, with $n_0 = 3$, the numbers $n_0 n_j$ are actually integers.

The differential operator $N_4$ satisfies the Calabi–Yau type conditions, and especially the series given in (G.5), (G.6), (G.7) and (G.9) have integer coefficients.

### G.2. The square root of the rational function of the LGF 4D body centered cubic

Let us consider the rational function $1/Q$, where the polynomial $Q$ reads:

$$ Q = 1 - (x+z) \cdot (1+y) \cdot (1+u) \cdot (1+v). \quad (G.12) $$

This is the multivariate polynomial corresponding to the Calabi–Yau number 3 (see table 1). The diagonal of $1/Q$ also identifies with the LGF of the 4D body centered cubic lattice [32].

The diagonal of $1/\sqrt{Q}$ with $Q$ given in (G.12), reads:

$$ \mathrm{Diag} \left( \frac{1}{\sqrt{Q}} \right) = {}_5F_4 \left( \left[ \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{4}, \frac{3}{4} \right], [1, 1, 1, 1], 256t \right). \quad (G.13) $$

This diagonal is annihilated by an irreducible order-five differential operator $L_5$, which has a differential Galois group included in the orthogonal group $SO(5, \mathbb{C})$. The operator $L_5$ has a $\ln(t)^4$ highest log-power formal solution at the origin. According to conjecture (3), a rational function $1/Q_{eq}$, depending on $6 = 4 + 2$ variables, should exist. It actually (with an extra variable $w$) reads $Q_{eq} = Q - w^{1/2}/4$. and the diagonal of $1/\sqrt{Q}$ actually identifies with the diagonal of $1/Q_{eq}$.

The order-five differential operator $L_5$ has MUM, and is actually the exterior square of an order-four differential operator, $L_5 = \mathrm{ext}^2(L_4)$. The differential operator $L_4$ has MUM, at $t = 0$, the indicial exponents being four times $1/2$.

Let us introduce the differential operator $N_4$:

$$ N_4 = L_4 \cdot \sqrt{t} \cdot (1 - 256t)^{1/4}. \quad (G.14) $$

With the scaling $t \to 16t$, the differential operator $N_4$ reads (with $\theta = tD_t$):

$$ \begin{aligned} N_4 = \theta^4 & - 16 \cdot t \cdot (758 \theta^4 + 512 \theta^3 + 440 \theta^2 + 184 \theta + 25) \\ & + 2^{12} \cdot t^2 \cdot (12288 \theta^4 + 16384 \theta^2 + 13056 \theta^3 + 3840 \theta + 401) \\ & - 2^{24} \cdot t^3 \cdot (8 \theta + 3)^2 \cdot (8 \theta + 5)^2. \end{aligned} \quad (G.15) $$

Similarly to the previous example, the differential operator $N_4$, given in (G.15), satisfies the Calabi–Yau type conditions.

## ORCID iD

J M Maillard https://orcid.org/0000-0002-8233-8501
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