# Comptes Rendus
Mathématique

Patricio Almirón and Mathias Schulze

Limit spectral distribution for non-degenerate hypersurface singularities

Volume 360 (2022), p. 699-710

<https://doi.org/10.5802/crmath.335>

© Académie des sciences, Paris and the authors, 2022.
Some rights reserved.

<figure><img src="image_1.png" /></figure>

This article is licensed under the
CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.
http://creativecommons.org/licenses/by/4.0/
K. Saito [16] was the first to study the asymptotic distribution of the spectrum. He considered the normalized spectrum of $f$,

$$ \chi_f(t) := \frac{\mathrm{Sp}_f(T)}{\mu} = \frac{1}{\mu} \sum_{j=1}^{\mu} T^{\alpha_j}, \quad T = \exp(2\pi i t), $$

as the Fourier transform of the discrete probability density on the interval $(0, n+1)$,

$$ \frac{1}{\mu} \sum_{i=1}^{\mu} \delta(s - \alpha_i)ds, $$

where $\delta(s)$ is Dirac's delta function. In the case of Brieskorn-Pham singularities (see Example 9) he identified a continuous limit probability distribution $N_{n+1}$ defined by

$$ N_{n+1}(s)ds := \int_{x_0+\cdots+x_n=s} \varphi(x_0)\cdots\varphi(x_n)dx_0\cdots dx_n, $$

where $\varphi$ is the indicator function of the unit interval $[0, 1]$,

$$ \varphi(x) := \begin{cases} 1 & \text{if } x \in [0, 1], \\ 0 & \text{if } x \notin [0, 1]. \end{cases} $$

Under the Fourier transform $\mathcal{F}$, $N_{n+1}$ corresponds to the power

$$ \mathcal{F}(N_{n+1}) = \mathcal{F}(\varphi)^{n+1}. \quad (1) $$

K. Saito [16, (2.5)i]] suggested to find singularities for which $\chi_f$ converges to $N_{n+1}$. Our main result establishes his limit spectral distribution for Newton non-degenerate singularities.

**Theorem 1.** For a fixed Newton diagram $\Gamma$, consider the Newton diagrams $\omega\Gamma$ obtained from $\Gamma$ by scaling with the factor $\omega$. Then we have

$$ \lim_{\omega \to \infty} \chi_{f_\omega} = \mathcal{F}(N_{n+1}), \quad (2) $$

where $f_\omega$ is any Newton non-degenerate function germ of $n+1$ variables with Newton diagram $\omega\Gamma$.

The proof is given in Section 4. Combined with the following remark, our result generates new cases where Saito's limit distribution is valid for a suitably chosen limit as in (2). The general choice of limit is unclear.

**Remark 2.** K. Saito [16, (3.7), (3.9), (3.2.1)] proved the following facts.

(a) For quasihomogeneous $f$ of degree 1 with respect to weights $w_0, \dots, w_n$, (2) holds, even with $\lim_{\omega \to \infty}$ replaced by $\lim_{w_0, \dots, w_n \to 0}$.

(b) For irreducible plane curve singularities $f$ with Puiseux pairs $(n_1, l_1), \dots, (n_g, l_g)$,

$$ \lim_{n_g \to \infty} \chi_f = \mathcal{F}(N_2). $$

(c) The join $f + g$ of two functions in disjoint sets of variables satisfies

$$ \chi_{f+g} = \chi_f \cdot \chi_g. $$

Therefore (2) is compatible with joins by (1).

K. Saito [16, (2.5)ii), (2.8)i]] further suggested to describe up to what extent the spectral distribution is bounded by $N_{n+1}$ and introduced the notion of (weakly) dominating values. Consider the function

$$ \Phi_f: [0, 1] \to \mathbb{R}, \quad r \mapsto \int_0^r N_{n+1}(s) - \frac{1}{\mu} \sum_{i=1}^{\mu} \delta(s - \alpha_i)ds $$

defined by the difference of the continuous and discrete spectral distributions. By definition $0 < r < \frac{n+1}{2}$ is a dominating value if $\Phi_f(r) > 0$ for all $f$ in $n+1$ variables. A weakly dominating value is defined by replacing $< by \le$ and $\int_0^r$ by $\int_0^{r-\epsilon}$ for all $\epsilon > 0$. In particular, K. Saito [16, (2.8)iv]]
formulated the following question which by work of M. Saito [17] extends Durfee's conjecture on the geometric genus

$$p_g = |\{i | \alpha_i \le 1\}| \qquad (3)$$

from the case of surface singularities.

**Question 3.** Is 1 a dominating value for all $n \ge 2$? In other words, for $f$ in $n+1$ variables, is the geometric genus bounded by

$$p_g < \frac{\mu}{(n+1)!}$$

Kerner and Nemethi [9] give a positive answer for Newton non-degenerate singularities with Newton diagram $\omega\Gamma$ for sufficiently large $\omega$.

As opposed to Question 3, Hertling's variance conjecture [6, Conj. 6.7] addresses the distribution of the spectrum in its full range.

**Conjecture 4 (Hertling's Variance Conjecture).** The variance of spectral numbers $\alpha_1 \le \dots \le \alpha_\mu$ is bounded by

$$\frac{1}{\mu} \sum_{i=1}^{\mu} \left( \alpha_i - \frac{n}{2} \right)^2 \le \frac{\alpha_\mu - \alpha_1}{12}.$$

It was confirmed by Brélivet [2, 3] for Newton non-degenerate singularities and for plane curves. We refer to Brélivet and Hertling [4] for refined investigations in this direction.

In Section 3, we investigate (the extremal) spectral numbers below 1 for their dominance in the case $n = 1$ of irreducible plane curve singularities $C = f^{-1}(0)$. For a single *Puiseux pair* $(p, q)$ we describe these spectral values in terms of the *value semigroup* $S = \langle p, q \rangle$ of $C$ (see (10)). This can be used to visualize the graph of $\Phi_f$ as a difference (see Figure 1).

<figure><img src="image_2.png" /><figcaption>**Figure 1.** The function $\Phi_f$ as a difference for $S = \langle 5, 9 \rangle$.</figcaption></figure>
One can write the smallest spectral number, the log canonical threshold, as $\frac{1}{p} + \frac{1}{q}$ and the largest below 1 as $1 - \frac{1}{pq}$. For these extremal spectral numbers we prove the following

**Proposition 5.** If $f(z_0, z_1)$ has a single Puiseux pair $(p, q)$, then

(a) $\Phi_f(\frac{1}{p} + \frac{1}{q}) > 0$ unless $p = 2$ and $q \in \{3, 5\}$, with $\lim_{p \to \infty} \Phi_f(\frac{1}{p} + \frac{1}{q}) = 0$,

(b) $\Phi_f(1 - \frac{1}{pq}) < 0$ with $\lim_{p \to \infty} \Phi_f(1 - \frac{1}{pq}) = 0$.

For general irreducible plane curve singularities, Igusa [8, Thm. 1] showed that the log canonical threshold depends only on multiplicity and the first Puiseux exponent (see also [14, Proof of Thm. 1.1]). It thus equals $\frac{1}{\bar{\beta}_0} + \frac{1}{\bar{\beta}_1}$ where $\bar{\beta}_0, \bar{\beta}_1$ are the two smallest minimal generators of the value semigroup. The statement of Proposition 5.(a) remains valid in this extended generality.

**Theorem 6.** For any irreducible plane curve singularity $C = f^{-1}(0)$ with value semigroup different from $\langle 2, 3 \rangle$ and $\langle 2, 5 \rangle$, we have $\Phi_f(\frac{1}{\bar{\beta}_0} + \frac{1}{\bar{\beta}_1}) > 0$. In other words, the squared log canonical threshold is bounded by

$$ \left( \frac{1}{\bar{\beta}_0} + \frac{1}{\bar{\beta}_1} \right)^2 > \frac{2}{\mu} $$

Moreover, $\lim_{n_g \to \infty} \Phi_f(\frac{1}{\bar{\beta}_0} + \frac{1}{\bar{\beta}_1}) = 0$.

In particular, Theorem 6 provides a quite surprising constraint on the first Puiseux pair of an irreducible plane curve singularity with a given Milnor number.

## Acknowledgements

The first named author wants to thank the second named author for his kindness and facilities for hosting him at TU Kaiserslautern in a pleasant working atmosphere during his research stay in September–November 2020 despite the difficulties of travels and face to face work due to the COVID19 pandemic.

## 2. Spectrum of non-degenerate singularities

Suppose that $f: (\mathbb{C}^{n+1}, 0) \to (\mathbb{C}, 0)$ is Newton non-degenerate. This means that there are local coordinates $z_0, \dots, z_n$ such that

$$ f = f(z_0, \dots, z_n) \in \mathbb{C}\{z_0, \dots, z_n\} = \mathcal{O}_{\mathbb{C}^{n+1}, 0} := \mathcal{O} $$

is a Newton non-degenerate convenient power series (see [11, 1.19 Def.] and [12, §II.8.5]). Let $\Gamma$ denote the Newton diagram of $f$. We write $\sigma \in \Gamma$ to indicate that $\sigma$ is a face of $\Gamma$. For $\sigma, \tau \in \Gamma$, write $\tau \le \sigma$ if $\tau$ is a face of $\sigma$. By $g_\sigma$ we denote the polynomial obtained from the power series $g \in \mathcal{O}$ by restricting the monomial support to the cone of $\sigma$.

There is a (decreasing) Newton filtration $\mathcal{N}$ defined by $\Gamma$ on $\mathcal{O}$. Following Steenbrink [19, (5.6)], denote the Newton graded ring associated to $\mathcal{O}$ by

$$ A := \mathrm{gr}_{\mathcal{N}} \mathcal{O} $$

and the principal parts of $z_0 \frac{\partial f}{\partial z_0}, \dots, z_n \frac{\partial f}{\partial z_n}$ with respect to $\mathcal{N}$ by $F_0, \dots, F_n$. For $\sigma \in \Gamma$ let $A_\sigma$ be the corresponding graded subring of $A$ and denote by

$$ d(\sigma) := \dim A_{\sigma} = \dim \mathcal{O}\sigma = \dim \sigma + 1 $$

its dimension.

The Brieskorn module (see [15, 18])

$$ \Omega_f := \Omega_{(\mathbb{C}^{n+1}, 0)}^{n+1} / \mathrm{d}f \wedge \Omega_{(\mathbb{C}^{n+1}, 0)}^n $$
also carries a Newton filtration which is induced by the inclusion

$$ \begin{array}{r@{\;}c@{\;}l@{\quad}l} \Omega_{(C^{n+1},0)}^{n+1} = & \mathcal{O} & \mathrm{d}z_0 \wedge \dots \wedge \mathrm{d}z_n \\ & \multicolumn{1}{@{}c@{}}{\xrightarrow{\substack{z_0 \cdots z_n \\ \mathrm{d}z_0 \wedge \dots \wedge \mathrm{d}z_n}}} & \mathcal{O} \\ \cline{2-3} & \multicolumn{1}{@{}c@{}}{\Big\downarrow} & \\ & \Omega_f & \mathcal{O} / \langle z_0 \frac{\partial f}{\partial z_0}, \dots, z_n \frac{\partial f}{\partial z_n} \rangle. \end{array} \quad (4) $$

M. Saito [18] and Varchenko-Khovanskiǐ [20] identified the Poincaré series of $\Omega_f$ with the singularity spectrum of $f$ defined by Steenbrink [19].

**Theorem 7 (M. Saito, Varchenko-Khovanskiǐ).** For Newton non-degenerate $f$, the Poincaré series of the Newton filtered vector space $\Omega_f$ reads

$$ p_{\Omega_f}(t) = t^{\alpha_1} + \dots + t^{\alpha_\mu} =: \mathrm{Sp}_f(t) $$

where $\alpha_1, \dots, \alpha_\mu$ are the spectral numbers of $f$.

□

The inclusion (4) identifies

$$ \begin{array}{r@{\;}c@{\;}l} \Omega_f & \cong & z_0 \cdots z_n \mathcal{O} / \left\langle z_0 \frac{\partial f}{\partial z_0}, \dots, z_n \frac{\partial f}{\partial z_n} \right\rangle, \\ \operatorname{gr}_{\mathcal{N}} \Omega_f & \cong & A / \langle F_0, \dots, F_n \rangle := H_f. \end{array} \quad (5) $$

Based on results of Kouchnirenko [11] (and Hochster [7]) Steenbrink [19, (5.7)] gave a formula for Newton non-degenerate $f$ decomposing $p_{H_f} = \mathrm{Sp}_f$ with respect to faces of the Newton diagram: For a face $\sigma \in \Gamma$ the first writes the Poincaré series of the subspace of $A_\sigma / \langle F_{0,\sigma}, \dots, F_{n,\sigma} \rangle$ corresponding to the interior of the cone $\mathbb{Q}_{\ge 0} \sigma$ of $\sigma$ as

$$ q_{\sigma}(t) = \sum_{\tau \le \sigma} (-1)^{d(\sigma) - d(\tau)} (1-t)^{d(\tau)} p_{A_{\sigma}}(t). $$

Denote the minimal dimension of a coordinate space containing $\sigma \in \Gamma$ by

$$ k(\sigma) := \min \{ k \in \mathbb{Z} \mid \exists i_1, \dots, i_k \in \{0, \dots, n\} : \sigma \subset \mathbb{Q}e_{i_1} + \dots + \mathbb{Q}e_{i_k} \}. $$

Then Steenbrink's formula is given by

**Theorem 8 (Steenbrink).** For Newton non-degenerate $f$ in $n+1$ variables, the Poincaré series of $H_f$ can be written as

$$ \begin{aligned} p_{H_f}(t) &= \sum_{\sigma \in \Gamma} (-1)^{n+1-d(\sigma)} (1-t)^{k(\sigma)} p_{A_{\sigma}}(t) \\ &= \sum_{\tau \le \sigma \in \Gamma} (-1)^{n+1-d(\sigma)} (1-t)^{k(\sigma)-d(\sigma)} q_{\sigma}(t). \end{aligned} \quad \square $$

### 3. Irreducible plane curve singularities

In this section, we elaborate on the case $n=1$ where $f$ defines an irreducible plane curve singularity $C = f^{-1}(0)$. We first consider the case of a single Puiseux pair and prove Proposition 5, then move on to the general case and prove Theorem 6.

Suppose first that $C$ has a single Puiseux pair $(p, q)$. Then $f$ is Newton non-degenerate with Newton diagram $\Gamma$ consisting of a single line segment $[(p,0), (q,0)]$ and defines an irreducible plane curve singularity $C = f^{-1}(0)$. The function $f$ is semiquasihomogeneous of weighted degree 1 with respect to weights

$$ w_0 = \frac{1}{p}, \quad w_1 = \frac{1}{q}, \quad d := pq, \quad (6) $$
on variables $z_0, z_1$ and can be written explicitly as

$$f(z_0, z_1) = z_0^p + z_1^q + \sum_{iq+jp>d} a_{i,j} z_0^i z_1^j.$$

The normalization $\tilde{C} \to C$ is given by

$$ \begin{array}{rcl} \mathcal{O}_C = \mathcal{O}/\langle f \rangle & \hookrightarrow & \mathcal{O}_{\tilde{C}} = \tilde{\mathcal{O}}_C \cong \mathbb{C}\{t\}, \\ \bar{z}_0 & \mapsto & t^q + \dots, \\ \bar{z}_1 & \mapsto & t^p + \dots. \end{array} \qquad (7) $$

The valuation $v: \tilde{\mathcal{O}}_C \to \mathbb{N}$, $v(t) = 1$, defines the *value semigroup*

$$S := v(\mathcal{O}_C \setminus \{0\}) = \langle p, q \rangle \subset \mathbb{N}.$$

Due to the finiteness of the normalization $S$ has a finite set of *gaps* $\mathbb{N} \setminus S$, which yields $k + \mathbb{N} \subset S$ for $k \gg 0$. The minimal such $k$ is the *conductor* of $S$ and equals the Milnor number (see [5, Prop. 1.2.1.1])

$$\mu = (p-1)(q-1). \qquad (8)$$

The Gorenstein property of $C$ is reflected by the symmetry between elements and gaps (see [13])

$$ \begin{array}{rcl} S & \xrightarrow{1:1} & \mathbb{Z} \setminus S, \\ a & \mapsto & \mu - 1 - a. \end{array} \qquad (9) $$

The normalized valuation $v/d$ induces the filtration $\mathcal{O}_C$ defined by weights $w = (w_0, w_1)$ on $z_0, z_1$. By assumption, this is the Newton filtration $\mathcal{N}$. Factoring (7) as

$$ \begin{array}{rcl} \mathcal{O}_C & \mathrel{\substack{\leftarrow\limits \\ \rightarrow\limits}} & \mathbb{C}\{t\} \\ \mathcal{O}/\langle z_0 \frac{\partial f}{\partial z_0}, z_1 \frac{\partial f}{\partial z_1} \rangle & \mathrel{\substack{\leftarrow\limits \\ \rightarrow\limits}} & \mathbb{C}\{t\}/\langle t^d \rangle \end{array} $$

and using (4) yields a Newton filtered inclusion

$$\mathcal{O}/\mathcal{N}_{1+w_0+w_1} \mathcal{O} \xrightarrow{dz_0 \wedge d\bar{z}_1} \Omega_f/\mathcal{N}_1 \Omega_f \hookrightarrow \mathbb{C}\{t\}/\langle t^d \rangle.$$

This identifies the corresponding ranges of spectral numbers and of values in the semigroup by means of

$$ \begin{array}{rcl} \{\alpha \in \{\alpha_1, \dots, \alpha_\mu\} \mid \alpha < 1 + w_0 + w_1\} & \xleftarrow{1:1} & S/\langle d \rangle, \\ \alpha & \mapsto & d\alpha - p - q, \\ \frac{k}{d} + w_0 + w_1 & \mapsto & k. \end{array} \qquad (10) $$

The smallest spectral number $w_0 + w_1$ corresponds to $0 \in S$, and the gap $\mu - 1$ of $S$ defining the Gorenstein symmetry (9) corresponds to the non-spectral number 1. It follows that (3) can be written explicitly as

$$p_g = \left| \left\{ (i, j) \in \mathbb{N}^2 \middle| \frac{i+1}{p} + \frac{j+1}{q} < 1 \right\} \right| = |\mathbb{N} \setminus S| = \frac{\mu}{2}. \qquad (11)$$

Under (9) the gap $1 \in \mathbb{N} \setminus S$ is the mirror of $\mu - 2 \in S$ and corresponds to the largest spectral number $1 - w_0 w_1 < 1$ by (10).

After these preparations we are ready to give the
**Proof of Proposition 5.**

(a). Using (6) and (8) we compute

$$
\begin{align*}
\mu\Phi_f(w_0 + w_1) &= \frac{\mu}{2}(w_0 + w_1)^2 - 1 = \frac{(p-1)(q-1)}{2} \left(\frac{1}{p} + \frac{1}{q}\right)^2 - 1 \\
&= \frac{(p-1)(q-1)(p+q)^2 - 2p^2q^2}{2p^2q^2} \\
&= \frac{(pq - p - q + 1)(p^2 + 2pq + q^2) - 2p^2q^2}{2p^2q^2} \tag{12} \\
&= \frac{2pq + p^3q - 3p^2q - p^3 + p^2 + pq^3 - 3pq^2 - q^3 + q^2}{2p^2q^2} \\
&= \frac{1}{pq} + \frac{pq - 3q - p + 1}{2q^2} + \frac{pq - 3p - q + 1}{2p^2},
\end{align*}
$$

which tends to 0 for $p \to \infty$. If $p \ge 4$ and $q \ge 5$, then (12) is positive since

$$
pq - 3q - p + 1 = p(q - 1) - 3q + 1 \geq 4q - 4 - 3q + 1 = q - 3 > 0,
$$

$$
pq - 3p - q + 1 = p(q - 3) - q + 1 \geq 4q - 12 - q + 1 \geq 3q - 11 > 0.
$$

If $p=3$, then (12) becomes

$$
\frac{1}{3q} - \frac{2}{2q^2} + \frac{2q-8}{18} = \frac{2q^3 - 8q^2 + 6q - 18}{18q^2},
$$

which is positive if $q \ge 4$. Finally, if $p=2$, then (12) becomes

$$
\frac{1}{2q} - \frac{q+1}{2q^2} + \frac{q-5}{8} = \frac{q^3 - 5q^2 - 4}{8q^2},
$$

which is positive if $q \ge 6$, but negative if $q \in \{3, 5\}$.

(b). Using (11) and (6) we compute

$$
\Phi_f(1 - w_0 w_1) = \frac{1}{2}(1 - w_0 w_1)^2 - \frac{1}{2} = -\frac{1}{2d^2} = -\frac{1}{2p^2q^2} < 0,
$$

which tends to 0 for $p \to \infty$.

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

Consider now an irreducible plane curve singularity $C = f^{-1}(0)$ with arbitrary number $g$ of
Puiseux pairs. To prepare the proof of Theorem 6, we review some standard integer invariants
(see [21, Ch. II, §1-3]): Let

$$
\bar{\beta}_0 < \bar{\beta}_1 < \dots < \bar{\beta}_g
$$

denote the minimal generators of the value semigroup of C and set

$$
e_i := \gcd(\bar{\beta}_0, \bar{\beta}_1, \dots, \bar{\beta}_i), \quad n_i := \frac{e_{i-1}}{e_i}, \quad q_i := \frac{\bar{\beta}_i}{e_i} \tag{13}
$$

for $i=0, \dots, g$. These greatest common divisors form a strictly decreasing sequence

$$
\bar{\beta}_0 = e_0 > e_1 > \dots > e_g = 1. \tag{14}
$$

Moreover, the minimal generators of the value semigroup satisfy inequalities

$$
n_{i-1} \bar{\beta}_{i-1} < \bar{\beta}_i \qquad (15)
$$

for $i=1, \dots, g$. The *characteristic Puiseux exponents* of $C$ are defined recursively by

$$
\beta_0 := \bar{\beta}_0, \quad \beta_1 := \bar{\beta}_1, \quad \beta_i := \bar{\beta}_i - n_{i-1}\bar{\beta}_{i-1} + \beta_{i-1}, \tag{16}
$$

for $i=2, \dots, g$. By (15) they form a strictly increasing sequence

$$
1 \le \beta_0 < \beta_1 < \dots < \beta_g. \tag{17}
$$
The Milnor number of $f$ can be written as (see [21, Ch. II, §3, (3.14)])

$$ \mu = \sum_{i=1}^{g} \beta_i (e_{i-1} - e_i) - \beta_0 + 1. \quad (18) $$

On the other hand, A'Campo showed that (see [1, Thm. 3.(ii)])

$$ \mu = \sum_{i=1}^{g} e_i \mu_i, \quad \mu_i := (n_i - 1)(q_i - 1). \quad (19) $$

**Proof of Theorem 6.** The case where $g = 1$ is covered by Proposition 5 (a). Using (14) and (17), we find a lower bound

$$ \begin{aligned} \mu &= -\beta_g e_g + \sum_{i=1}^{g-1} (\beta_{i+1} - \beta_i) e_i + \beta_1 e_0 - \beta_0 + 1 \\ &\geq -\beta_g e_g + \beta_1 e_0 - \beta_0 + 1 \\ &= -\beta_g + \beta_0(\beta_1 - 1) + 1 \\ &> -\beta_g + \beta_0 \beta_1 - \beta_0. \end{aligned} \quad (20) $$

Suppose first that $g \ge 3$. Using (18) and (20), we compute

$$ \begin{aligned} (\beta_0 + \beta_1)^2 \mu - 2\beta_0^2 \beta_1^2 &= (\beta_0^2 + 2\beta_0\beta_1 + \beta_1^2)\mu - 2\beta_0^2 \beta_1^2 \\ &> \sum_{i=1}^{g} \beta_0^2 \beta_i (e_{i-1} - e_i) - \beta_0^3 + \beta_0^2 \\ &\quad + \sum_{i=1}^{g} \beta_1^2 \beta_i (e_{i-1} - e_i) - \beta_0 \beta_1^2 + \beta_1^2 \\ &\quad - 2\beta_0 \beta_1 \beta_g - 2\beta_0^2 \beta_1 \\ &> \sum_{i=1}^{g-1} \beta_0^2 \beta_i + \sum_{i=1}^{g-1} \beta_1^2 \beta_i - \beta_0^3 - \beta_0 \beta_1^2 \\ &\quad + (\beta_0 - \beta_1)^2 \beta_g - 2\beta_0^2 \beta_1 \\ &> 2\beta_0^2 \beta_1 + 2\beta_1^3 - \beta_0^3 - \beta_0 \beta_1^2 - 2\beta_0^2 \beta_1 > 0. \end{aligned} $$

It follows that

$$ \Phi_f \left( \frac{1}{\bar{\beta}_0} + \frac{1}{\bar{\beta}_1} \right) = \frac{1}{2} \left( \frac{1}{\bar{\beta}_0} + \frac{1}{\bar{\beta}_1} \right)^2 - \frac{1}{\mu} = \frac{(\beta_0 + \beta_1)^2 \mu - 2\beta_0^2 \beta_1^2}{2\beta_0^2 \beta_1^2 \mu} > 0. $$

Suppose now that $g = 2$. By (19), (13), (14) and (15),

$$ e_1 \mu_1 = (n_1 - 1)(\bar{\beta}_1 - e_1) = n_1 \bar{\beta}_1 - \bar{\beta}_1 - e_0 + e_1 < n_1 \bar{\beta}_1 \le \bar{\beta}_2 - 1 = q_2 - 1 $$

and hence

$$ \begin{aligned} \mu - e_1^2 \mu_1 &= e_1 \mu_1 + e_2 \mu_2 - e_1^2 \mu_1 \\ &> e_1 (1 + e_2(n_2 - 1) - e_1) \mu_1 \\ &= e_1 (1 + e_1 - e_2 - e_1) \mu_1 \\ &= e_1 (1 - e_2) \mu_1 = 0. \end{aligned} \quad (21) $$

If $(n_1, q_1) \notin \{(2,3), (2,5)\}$, then by (13), Proposition 5 and (21)

$$ \Phi_f \left( \frac{1}{\bar{\beta}_0} + \frac{1}{\bar{\beta}_1} \right) = \frac{1}{e_1^2} \frac{1}{2} \left( \frac{1}{n_1} + \frac{1}{q_1} \right)^2 - \frac{1}{\mu} > \frac{1}{e_1^2 \mu_1} - \frac{1}{\mu} = \frac{\mu - e_1^2 \mu_1}{e_1^2 \mu_1 \mu} > 0. $$
Otherwise, we have $n_1 = 2 \le e_1 = n_2$ and (15) yields $q_2 > 2e_1q_1$. Using (19) it follows that

$$
\begin{align*}
\mu &= (e_1 - 1)(q_2 - 1) + e_1(q_1 - 1) \\
&> (e_1 - 1)(2e_1q_1 - 1) + e_1(q_1 - 1) \\
&= 2e_1q_1(e_1 - 1) + e_1(q_1 - 2) + 1 \\
&= 2q_1e_1^2 - (q_1 + 2)e_1 + 1
\end{align*}
$$

and hence

$$
(2+q_1)^2 \mu - 8e_1^2 q_1^2 = \begin{cases} 78e_1^2 - 125e_1 + 25 & \text{if } q=3, \\ 290e_1^2 - 343e_1 + 49 & \text{if } q=5. \end{cases}
$$

In both cases $e_1 \ge 2$ implies

$$
\Phi_f \left( \frac{1}{\beta_0} + \frac{1}{\beta_1} \right) = \frac{(n_1 + q_1)^2 \mu - 2e_1^2 n_1^2 q_1^2}{2e_1^2 n_1^2 q_1^2 \mu} = \frac{(2 + q_1)^2 \mu - 8e_1^2 q_1^2}{8e_1^2 q_1^2 \mu} > 0
$$

which tend to 0 for $n_g \to \infty$ since this entails $e_1 \to \infty$ by (14) and $\mu \to \infty$ by (19)

<figure><img src="image_4.png" /></figure>

4. Limit spectral distribution

In this section we return to the general setup of Section 2 and prove our main result Theorem 1.
Our approach is to subdivide the Newton diagram and mimic an argument of K. Saito (see [16,
(2.2), (3.7)]). We begin with his motivating

**Example 9 (Brieskorn–Pham type singularities).** Suppose first that $n=0$ and $f=f(z)=z^d$ is quasihomogeneous of degree 1 with respect to the weight $w=1/d$ on $z$ with Milnor number $\mu=d-1$. By (5), $H=\langle z \rangle \subset \mathbb{C}\{z\}/\langle z^d \rangle$ and hence

$$
p_{H_f}(t) = \frac{t(t^{w-1} - 1)}{1 - t^w} = \frac{t - t^w}{t^w - 1}.
$$

By Theorem 8, using L'Hôpital's rule in the second step,

$$
\begin{align*}
\lim_{w \to 0} \chi_f(t) &= \lim_{w \to 0} \frac{p_H(T)}{\mu} \tag{22} \\
&= \lim_{w \to 0} \frac{w \exp(2\pi it) - \exp(2\pi i t w)}{1-w \exp(2\pi i t w) - 1} \\
&= \lim_{w \to 0} \frac{\exp(2\pi it) - \exp(2\pi i t w) - 2\pi i t w \exp(2\pi i t w)}{1 - \exp(2\pi i t w) + 2\pi i (1-w) \exp(2\pi i t w)} \\
&= \frac{\exp(2\pi i t) - 1}{2\pi i t} = \frac{\exp(\pi i t) \exp(\pi i t) - \exp(-\pi i t)}{2i} \\
&= \frac{\exp(\pi i t)}{\pi t} \sin(\pi t) = \mathcal{F}(\varphi)(t).
\end{align*}
$$

Consider now $f = f_0 + \cdots + f_n$, where $f_j = f_j(z_j) = z_j^{d_j}$, which is quasihomogeneous of degree 1 with respect to weights $w_0 = 1/d_0, \dots, w_n = 1/d_n$ on the variables $z_0, \dots, z_n$ with Milnor number $\mu = \mu_f = \prod_{j=0}^n \mu_{f_j}$. Then $H = H_f = H_{f_0} \otimes_C \cdots \otimes_C H_{f_n}$ and hence, by the first part and (1),

$$
\lim_{w_0, \dots, w_n \to 0} \chi_f(t) = \prod_{i=0}^{n} \lim_{w_i \to 0} \chi_{f_i}(t) = \mathcal{F}(\varphi)^{n+1}(t) = \mathcal{F}(N_{n+1})(t).
$$

In this sense the normalized spectrum converges in distribution to the continuous probability
distribution $N_{n+1}$.

For our purpose we adapt the calculation (22) as follows.

**Lemma 10.** $\lim_{w \to 0} w^{\frac{1-T}{1-Tw}} = \mathcal{F}(\varphi)(t)$.
**Proof.** Using L'Hôpital's rule in the second step and (22), we compute

$$ 
\begin{aligned}
\lim_{w \to 0} w \frac{1-T}{1-T^w} &= \lim_{w \to 0} \frac{w \cdot (1 - \exp(2\pi i t))}{1 - \exp(2\pi i t w)} \\
&= \lim_{w \to 0} \frac{1 - \exp(2\pi i t)}{-2\pi i t \exp(2\pi i t w)} \\
&= \frac{1 - \exp(2\pi i t)}{-2\pi i t} = \mathcal{F}(\varphi)(t).
\end{aligned}
\quad \square $$

For the subdivision of the Newton diagram we rely on the following general result. The basis of a rational pointed cone $\sigma$ are the irreducible integral vectors $\alpha_0, \dots, \alpha_k$ on its rays. If it extends to a lattice basis, then $\sigma$ is called *regular*. In this case $\sigma$ is a simplicial cone and the convex hull of $\{0, \alpha_0, \dots, \alpha_k\}$ has $k$-dimensional volume 1 (see [10, §1.1]). A rational fan is called *regular* if all its cones are regular. Varchenko [10, Thm. 1, Remark] pointed out the following

**Theorem 11.** Any finite rational fan has a regular subdivision. $\square$

Finally, we make use of Kouchnirenko's formula [11, Thm. I] for the Milnor number of Newton non-degenerate singularities in terms of volumes.

**Theorem 12 (Kouchnirenko).** The Milnor number of any Newton non-degenerate $f$ in $n+1$ variables can be written as

$$ \mu = (n+1)!V_{n+1} - n!V_n + \dots + 1!(-1)^n V_1 + (-1)^{n+1}, $$

where $V_k$ is the sum of $k$-dimensional volumes of the intersection of the convex hull of $\Gamma \cup \{0\}$ with the $k$-dimensional coordinate planes. $\square$

We are now ready for the

**Proof of Theorem 1.** By Theorem 11, $\Gamma$ has a subdivision $\tilde{\Gamma}$ corresponding to a regular subdivision of its fan of cones. For any $\tau \in \tilde{\Gamma}$ let $w_0^\tau, \dots, w_k^\tau$ be the weights of the basis of $\mathbb{Q}_{\ge 0} \tau \cap \mathbb{Z}$. Then

$$ p_{A_\sigma}(t) = \sum_{\tilde{\Gamma} \ni \tau \le \sigma} (-1)^{d(\sigma)-d(\tau)} p_{A_\tau}(t), \quad p_{A_\tau}(t) = \prod_{j=0}^{\dim \tau} \frac{1}{1 - t^{w_j^\tau}}. $$

Substituting into Steenbrink's formula from Theorem 8 yields

$$ p_{H_f}(t) = \sum_{\tilde{\Gamma} \ni \tau \le \sigma \in \Gamma} (-1)^{n+1-d(\tau)} \frac{(1-t)^{k(\sigma)}}{\prod_{j=0}^{\dim \tau} (1-t^{w_j^\tau})}. $$

Passing to $\tilde{\omega}\Gamma$, $w_j^\tau$ is preplaced by $\varepsilon w_j^\tau$ where $\varepsilon \tilde{\omega} = 1$ and hence

$$ p_{H_{f_{\tilde{\omega}}}}(t) = \sum_{\tilde{\Gamma} \ni \tau \le \sigma \in \Gamma} (-1)^{n+1-d(\tau)} \frac{(1-t)^{k(\sigma)}}{\prod_{j=0}^{\dim \tau} (1-t^{\varepsilon w_j^\tau})}. $$

By Theorems 7 and 12,

$$ \lim_{\tilde{\omega} \to \infty} \chi_{f_{\tilde{\omega}}}(t) = \lim_{\tilde{\omega} \to \infty} \frac{p_{H_{f_{\tilde{\omega}}}}(T)}{\mu_{f_{\tilde{\omega}}}} \quad (23) $$

$$ = \sum_{\tilde{\Gamma} \ni \tau \le \sigma \in \Gamma} (-1)^{n+1-d(\tau)} \lim_{\tilde{\omega} \to \infty} \frac{1}{\mu_{f_{\tilde{\omega}}}} \frac{(1-t)^{k(\sigma)}}{\prod_{j=0}^{\dim \tau} (1-t^{\varepsilon w_j^\tau})}, $$

$$ \mu_{f_{\tilde{\omega}}} = \sum_{j=0}^{n+1} (-1)^{n+1-j} j! \tilde{\omega}^j V_j. \quad (24) $$
Fix $\tilde{\Gamma} \ni \tau \le \sigma \in \Gamma$. Let $V(\tau)$ be the $d(\tau)$-dimensional volume of the convex hull of $\tau \cup \{0\}$. Note that

$$ \sum_{\substack{\tau \in \tilde{\Gamma} \\ d(\tau) = n+1}} V(\tau) = V_{n+1}, \qquad (25) $$

$$ 1/V(\tau) = d(\tau)! \prod_{j=0}^{d(\tau)} w_j^{\tau}. \qquad (26) $$

The summand in (23) indexed by $\tau$ is then computed using (26), Lemma 10 and (24):

$$ \begin{aligned} & \lim_{\substack{\omega \to \infty}} \frac{1}{\mu_{f_{\omega}}} \frac{(1-T)^{k(\sigma)}}{\prod_{j=0}^{\dim \tau} (1-T^{\epsilon} w_j^{\tau})} \\ &= \lim_{\substack{\omega \to \infty}} \frac{\prod_{j=0}^{\dim \tau} \frac{\omega}{w_j^{\tau}}}{\mu_{f_{\omega}}} (1-T)^{k(\sigma)-d(\tau)} \prod_{j=0}^{\dim \tau} \lim_{\epsilon \to 0} \epsilon w_j^{\tau} \frac{1-T}{1-T^{\epsilon} w_j^{\tau}} \\ &= \lim_{\substack{\omega \to \infty}} \frac{d(\tau)! V(\tau) \omega^{d(\tau)}}{\mu_{f_{\omega}}} (1-T)^{k(\sigma)-d(\tau)} \left( \lim_{w \to 0} w \frac{1-T}{1-T^w} \right)^{d(\tau)} \\ &= \lim_{\substack{\omega \to \infty}} \frac{d(\tau)! V(\tau) \omega^{d(\tau)}}{\mu_{f_{\omega}}} (1-T)^{k(\sigma)-d(\tau)} \mathcal{F}(N_{d(\tau)})(t) \\ &= \begin{cases} \frac{V(\tau)}{V_{n+1}} \mathcal{F}(N_{n+1})(t) & \text{if } d(\tau) = n+1, \\ 0 & \text{if } d(\tau) < n+1. \end{cases} \end{aligned} $$

The claim now follows by substituting into (23) and applying (25). $\square$

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