pinecone/core-2020-05-10-deduplication huggingface dataset:
Electronic Communication of Mathematics and the Interaction of Computer Algebra Systems and Proof Assistants

AbstractPresent day computer algebra systems (CASs) and proof assistants (PAs) are specialized programs that help humans with mathematical computations and deductions. Although several such systems are impressive, they all have certain limitations. In most CASs side conditions that are essential for the truth of an equality are not formulated; moreover there are bugs. The PAs have a limited power for computing and hence also for assistance with proofs. Almost all examples of both categories are stand alone special purpose systems and therefore they cannot communicate with each other. We will argue that the present state of the art in logic is such that there is a natural formal language, independent of the special purpose application in question, by which these systems can communicate mathematical statements. In this way their individual power will be enhanced. Statements received at one particular location from other sites fall into two categories: with or without the qualification “evidently impeccable", a notion that is methodologically precise and sound. For statements having this quality assessment the evidence may come from the other site or from the local site itself, but in both cases it is verified locally. In cases where there is no evidence of impeccability one has to rely on cross checking. There is a trade-off between these two kinds of statements: for impeccability one has to pay the price of obtaining less power. Some examples of communication forms are given that show how the participants benefit
Commutative Algebra in the Mizar System

AbstractWe report on the development of algebra in the Mizar system. This includes the construction of formal multivariate power series and polynomials as well as the definition of ideals up to a proof of the Hilbert basis theorem. We present how the algebraic structures are handled and how we inherited the past developments from the Mizar Mathematical Library (MML). The MML evolves and past contributions are revised and generalized. Our work on formal power series caused a number of such revisions. It seems that revising past developments with an intent to generalize them is a necessity when building a database of formalized mathematics. This poses a question: how much generalization is best
Formal and Efficient Primality Proofs by Use of Computer Algebra Oracles

AbstractThis paper focuses on how to use Pocklington’s criterion to produce efficient formal proof-objects for showing primality of large positive numbers. First, we describe a formal development of Pocklington’s criterion, done using the proof assistant Coq. Then we present an algorithm in which computer algebra software is employed as oracle to the proof assistant to generate the necessary witnesses for applying the criterion. Finally, we discuss the implementation of this approach and tackle the proof of primality for some of the largest numbers expressible in Coq
Rational Solutions of Riccati-like Partial Differential Equations

AbstractWhen factoring linear partial differential systems with a finite-dimensional solution space or analysing symmetries of nonlinear ODEs, we need to look for rational solutions of certain nonlinear PDEs. The nonlinear PDEs are called Riccati-like because they arise in a similar way as Riccati ODEs. In this paper we describe the structure of rational solutions of a Riccati-like system, and an algorithm for computing them. The algorithm is also applicable to finding all rational solutions of Lie’s system { ∂xu+u2+a1u+a2v+a3, ∂yu+uv+b1u+b2v+b3, ∂xv+uv+c1u+c2v+c3, ∂yv+v2+d1u+d2v+d3},where a1, . . . , d3are rational functions of x and y
Fast Computation of the Bezout and Dixon Resultant Matrices

AbstractEfficient algorithms are derived for computing the entries of the Bezout resultant matrix for two univariate polynomials of degree n and for calculating the entries of the Dixon–Cayley resultant matrix for three bivariate polynomials of bidegree (m, n). Standard methods based on explicit formulas requireO (n3) additions and multiplications to compute all the entries of the Bezout resultant matrix. Here we present a new recursive algorithm for computing these entries that uses onlyO (n2) additions and multiplications. The improvement is even more dramatic in the bivariate setting. Established techniques based on explicit formulas requireO (m4n4) additions and multiplications to calculate all the entries of the Dixon–Cayley resultant matrix. In contrast, our recursive algorithm for computing these entries uses onlyO (m2n3) additions and multiplications
Improved Projection for Cylindrical Algebraic Decomposition

AbstractMcCallum’s projection operator for cylindrical algebraic decomposition (CAD) represented a huge step forward for the practical utility of the CAD algorithm. This paper presents a simple theorem showing that the mathematics in McCallum’s paper actually point to a better projection operator than he proposes—a reduced McCallum projection. The reduced projection has the potential to not simply speed up CAD computation for problems that are currently solvable in practice, but actually increase the scope of problems that can realistically be attacked via CADs. Additionally, the same methods are used to show that McCallum’s projection can be reduced still further when CAD is applied to certain types of commonly occurring quantifier elimination problems
Counting Points on Curves and Abelian Varieties Over Finite Fields

AbstractWe develop efficient methods for deterministic computations with semi-algebraic sets and apply them to the problem of counting points on curves and Abelian varieties over finite fields. For Abelian varieties of dimension g in projective N space overFq, we improve Pila’s result and show that the problem can be solved in O((logq)δ) time where δ is polynomial in g as well as in N. For hyperelliptic curves of genus g overFq we show that the number of rational points on the curve and the number of rational points on its Jacobian can be computed in (logq)O(g2logg)time
Squarefree Decomposition of Univariate Polynomials Depending on a Parameter. Application to the Integration of Parametric Rational Functions

AbstractIn this paper, we describe the application of a new version of Barnett’s method to the squarefree decomposition of a univariate polynomial with coefficients in K [ x ], x being a parameter andK a characteristic zero field. This new version of Barnett’s method uses Bezoutian matrices instead of matrices obtained from evaluating polynomials in a companion matrix and allows the determination of the squarefree decomposition parametrizing the gcd of the polynomial and its successive derivatives with respect to the main variable. The application of this parametric squarefree decomposition to the integration of parametric rational functions is also presented
Symbolic Reachability Computation for Families of Linear Vector Fields

AbstractThe control paradigm of physical processes being supervised by digital programs has lead to the development of a theory of hybrid systems combining finite state automata with differential equations. One of the most important problems in the verification of hybrid systems is the reachability problem. Even though the computation of reachable spaces for finite state machines is well developed, computing the reachable space of a differential equation is difficult. In this paper, we present the first known families of linear differential equations with a decidable reachability problem. This is achieved by posing the reachability computation as a quantifier elimination problem in the decidable theory of the reals. We illustrate the applicability of our approach by performing computations using the packages Redlog and Qepcad. Such symbolic computations can be incorporated in computer-aided verification tools for purely discrete systems, resulting in verification tools for hybrid systems with linear differential equations
Multivariate Polynomial System Solving Using Intersections of Eigenspaces

AbstractThe solutions of a polynomial system can be computed using eigenvalues and eigenvectors of certain endomorphisms. There are two different approaches, one by using the (right) eigenvectors of the representation matrices, one by using the (right) eigenvectors of their transposed ones, i.e. their left eigenvectors. For both approaches, we describe the common eigenspaces and give an algorithm for computing the solution of the algebraic system. As a byproduct, we present a new method for computing radicals of zero-dimensional ideals
Computing Projectors, Injectors, Residuals and Radicals of Finite Soluble Groups

AbstractLet G be a finite soluble group, given by a polycyclic generating system, and X a class of groups, represented by an algorithm that decides whether a given finite group belongs to X or not. This paper contains practical algorithms for the computation of X -projectors and X -injectors of G, whereX is an arbitrary Schunck class or Fitting class, respectively. It also describes algorithms for computing X -radicals for Fitting classes X and X -residuals for formations X. Variants of the latter algorithms can be used to compute all normal (X -)subgroups and the socle of G
Minimal Free Resolutions of HomogenizedD-modules

AbstractHomogenizing a module over the ring of differential operators, we define the notion of a minimal free resolution that is adapted to a filtration. We show that one can apply a modification of the algorithm of La Scala and Stillman to compute such a free resolution. By dehomogenization, one gets a free resolution of the original module that is small enough to compute, e.g. its restriction and integration. We have implemented our algorithm in a computer algebra system Kan and give examples by using this implementation
Constructibility of the Set of Polynomials with a Fixed Bernstein–Sato Polynomial: an Algorithmic Approach

AbstractLetnanddbe positive integers, letkbe a field and letP(n,d ;k)be the space of the non-zero polynomials innvariables of degree at mostdwith coefficients ink. LetB(n, d)be the set of the Bernstein–Sato polynomials of all polynomials inP(n,d ;k)askvaries over all fields of characteristic0. G. Lyubeznik proved thatB(n, d)is a finite set and asked if, for a fixedk, the set of the polynomials corresponding to each element ofB(n, d)is a constructible subset ofP(n, d;k).In this paper we give an affirmative answer to Lyubeznik’s question by showing that the set in question is indeed constructible and defined overQ, i.e. its defining equations are the same for all fieldsk. Moreover, we construct an algorithm that for each pair(n,d )produces a complete list of the elements ofB(n,d )and, for each element of this list, an explicit description of the constructible set of polynomials having this particular Bernstein–Sato polynomial
Introduction to the GiNaC Framework for Symbolic Computation within the C++ Programming Language

AbstractThe traditional split into a low level language and a high level language in the design of computer algebra systems may become obsolete with the advent of more versatile computer languages. We describe GiNaC, a special-purpose system that deliberately denies the need for such a distinction. It is entirely written in C++and the user can interact with it directly in that language. It was designed to provide efficient handling of multivariate polynomials, algebras and special functions that are needed for loop calculations in theoretical quantum field theory. It also bears some potential to become a more general purpose symbolic package
Local Box Adjacency Algorithms for Cylindrical Algebraic Decompositions

AbstractWe describe new algorithms for determining the adjacencies between zero-dimensional cells and those one-dimensional cells that are sections (not sectors) in cylindrical algebraic decompositions (cad). Such adjacencies constitute a basis for determining all other cell adjacencies. Our new algorithms are local, being applicable to a specified 0D cell and the 1D cells described by specified polynomials. Particularly efficient algorithms are given for the 0D cells in spaces of dimensions two, three and four. Then an algorithm is given for a space of arbitrary dimension. This algorithm may on occasion report failure, but it can then be repeated with a modified isolating interval and a likelihood of success
Gröbner Bases in Orders of Algebraic Number Fields

AbstractWe prove that any orderOof any algebraic number field K is a reduction ring. Rather than showing the axioms for a reduction ring hold, we start from scratch by well-orderingO, defining a division algorithm, and demonstrating how to use it in a Buchberger algorithm which computes a Gröbner basis given a finite generating set for an ideal. It is shown that our theory of Gröbner bases is equivalent to the ideal membership problem and in fact, a total of eight characterizations are given for a Gröbner basis. Additional conclusions and questions for further investigation are revealed at the end of the paper
A New Algorithm for Discussing Gröbner Bases with Parameters

AbstractLet F be a set of polynomials in the variables __x=x1, . . . , xnwith coefficients in R [__ a ], where R is a UFD and __ a=a1, . . . ,am a set of parameters. In this paper we present a new algorithm for discussing Gröbner bases with parameters. The algorithm obtains all the cases over the parameters leading to different reduced Gröbner basis, when the parameters in F are substituted in an extension field K of R. This new algorithm improves Weispfenning’s comprehensive Gröbner basis CGB algorithm, obtaining a reduced complete set of compatible and disjoint cases. A final improvement determines the minimal singular variety outside of which the Gröbner basis of the generic case specializes properly. These constructive methods provide a very satisfactory discussion and rich geometrical interpretation in the applications
Approximate Solutions of Polynomial Equations

AbstractIn this paper, we introduce “approximate solutions" to solve the following problem: given a polynomial F(x, y) over Q, where x represents an n -tuple of variables, can we find all the polynomials G(x) such that F(x, G(x)) is identically equal to a constant c in Q ? We have the following: let F(x, y) be a polynomial over Q and the degree of y in F(x, y) be n. Either there is a unique polynomial g(x) ∈Q [ x ], with its constant term equal to 0, such that F(x, y) =∑j=0ncj(y−g(x))jfor some rational numbers cj, hence, F(x, g(x) +a) ∈Q for all a∈Q, or there are at most t distinct polynomials g1(x),⋯ , gt(x), t≤n, such that F(x, gi(x)) ∈Q for 1 ≤i≤t. Suppose that F(x, y) is a polynomial of two variables. The polynomial g(x) for the first case, or g1(x),⋯ , gt(x) for the second case, are approximate solutions of F(x, y), respectively. There is also a polynomial time algorithm to find all of these approximate solutions. We then use Kronecker’s substitution to solve the case of F(x, y)
The Riccati Equation in Mathematical Finance

AbstractThis paper uses ideas from symbolic computation to classify solutions to an important class of problems in mathematical finance and thus provides a linkage between these two fields. We show that Kovacic’s concept of closed-form solutions to the Riccati ordinary differential equation can be used to provide a precise mathematical definition that is useful in certain financial models. We extend this definition to a broader class of problems and discuss how these ideas can be usefully applied to practical problems in the finance area. We provide a specific application by developing a new implementation of the Cox–Ingersoll–Ross interest-rate model that may be of practical interest
Size Reduction and Partial Decoupling of Systems of Equations

AbstractA method is presented that reduces the number of terms of systems of linear equations (algebraic, ordinary and partial differential equations). As a byproduct these systems have a tendency to become partially decoupled and are more likely to be factorizable or integrable. A variation of this method is applicable to nonlinear systems. Modifications to improve efficiency are given and examples are shown. This procedure can be used in connection with the computation of the radical of a differential ideal (differential Gröbner basis)
Identities for the Associator in Alternative Algebras

AbstractThe associator is an alternating trilinear product for any alternative algebra. We study this trilinear product in three related algebras: the associator in a free alternative algebra, the associator in the Cayley algebra, and the ternary cross product on four-dimensional space. This last example is isomorphic to the ternary subalgebra of the Cayley algebra which is spanned by the non-quaternion basis elements. We determine the identities of degree ≤ 7 satisfied by these three ternary algebras. We discover two new identities in degree 7 satisfied by the associator in every alternative algebra and five new identities in degree 7 satisfied by the associator in the Cayley algebra. For the ternary cross product we recover the ternary derivation identity in degree 5 introduced by Filippov
A Family of Sparse Polynomial Systems Arising in Chemical Reaction Systems

AbstractThe positive steady states of chemical reaction systems modeled by mass action kinetics are investigated. This sparse polynomial system is given by a weighted directed graph and a weighted bipartite graph. In this application the number of real positive solutions within certain affine subspaces ofRmis of particular interest. We show that the simplest cases are equivalent to binomial systems and are explained with the help of toric varieties. The argumentation is constructive and suggests algorithms. In general the solution structure is highly determined by the properties of the two graphs. We explain how the graphs determine the Newton polytopes of the system of sparse polynomials and thus determine the solution structure. Results on positive solutions from real algebraic geometry are applied to this particular situation. Examples illustrate the theoretical results
Identifying Half-Twists Using Randomized Algorithm Methods

AbstractSince the braid group was discovered by Artin (1947), the question of its conjugacy problem has been solved by Garside (1969) and Birman et al.(1998). However, the solutions given thus far are difficult to compute with a computer, since the number of operations needed is extremely large. Meanwhile, random algorithms used to solve difficult problems such as the primality of a number were developed, and the random practical methods have become an important tool. We give a random algorithm, along with a conjecture of how to improve its convergence speed, in order to identify elements in the braid group, which are conjugated to its generators (sayσ1k ) for a given power k. These elements of the braid group, the half-twists, are important in themselves, as they are the key players in some geometrical and algebraical methods, the building blocks of quasipositive braids and they construct endless sets of generators for the group
Rational Normal Forms and Minimal Decompositions of Hypergeometric Terms

AbstractWe describe a multiplicative normal form for rational functions which exhibits the shift structure of the factors, and investigate its properties. On the basis of this form we propose an algorithm which, given a rational function R, extracts a rational part F from the product of consecutive values of R: ∏k=n0n−1R(k) =F(n)∏k=n0n−1V(k) where the numerator and denominator of the rational function V have minimal possible degrees. This gives a minimal multiplicative representation of the hypergeometric term ∏k=n0n−1R(k). We also present an algorithm which, given a hypergeometric term T(n), constructs hypergeometric terms T1(n) and T2(n) such that T(n) =ΔT1(n) +T2(n) and T2(n) is minimal in some sense. This solves the additive decomposition problem for indefinite sums of hypergeometric terms: ΔT1(n) is the “summable part", and T2(n) the “non-summable part" of T(n). In other words, we get a minimal additive decomposition of the hypergeometric term T(n)
Scalable Parallel Coset Enumeration: Bulk Definition and the Memory Wall

AbstractCoset enumeration, like Gröbner bases, is a notoriously difficult algorithm to parallelize. We demonstrate a successful shared memory parallelization achieving a seven times speedup on an Origin 2000 CC-NUMA computer using 16 CPUs. We take as a testbed, an enumeration of Lyons’s group (8835156 cosets). This provides comparability with previous efforts in the literature (Cooperman and Havas, 1997; Havas and Ramsay, 2000) for which the best previous speedup was a factor of 4. The new parallelization depends on two new heuristics, clouds and shallow scan. Clouds is an example of bulk definition of cosets, which forms the key to our more efficient parallelization. The parallelization is implemented using TOP-C. By taking advantage of TOP-C’s option to compile for either shared or distributed memory, we also demonstrate the first efficient parallelization of a coset enumeration program using distributed memory.Our faster results expose for the first time in the context of coset enumeration the “memory wall", i.e. the latency barrier of the RAM. We verify this memory wall by showing on an Origin 2000 that a memory-bound parallel program with 64 CPUs doing nothing but randomly accessing RAM achieves at best a speedup of only 3.75 over the single-CPU version. Further, it is demonstrated that even sequential versions of coset enumerations programs are memory latency-bound in today’s technology. The lessons of bulk definition and of the memory wall carry over to related algorithms such as Gröbner bases, Knuth-Bendix, and other symbolic algebra algorithms with intermediate swell
Hybrid Sparse Resultant Matrices for Bivariate Polynomials

AbstractWe study systems of three bivariate polynomials whose Newton polygons are scaled copies of a single polygon. Our main contribution is to construct square resultant matrices, which are submatrices of those introduced by Cattaniet al. (1998), and whose determinants are nontrivial multiples of the sparse (or toric) resultant. The matrix is hybrid in that it contains a submatrix of Sylvester type and an additional row expressing the toric Jacobian. If we restrict our attention to matrices of (almost) Sylvester-type and systems as specified above, then the algorithm yields the smallest possible matrix in general. This is achieved by strongly exploiting the combinatorics of sparse elimination, namely by a new piecewise-linear lifting. The major motivation comes from systems encountered in geometric modeling. Our preliminary Maple implementation, applied to certain examples, illustrates our construction and compares it with alternative matrices
Derivations and Radicals of Polynomial Ideals over Fields of Arbitrary Characteristic

AbstractThe purpose of this paper is to give a complete effective solution to the problem of computing radicals of polynomial ideals over general fields of arbitrary characteristic. We prove that Seidenberg’s “Condition P" is both a necessary and sufficient property of the coefficient field in order to be able to perform this computation. Since Condition P is an expensive additional requirement on the ground field, we use derivations and ideal quotients to recover as much of the radical as possible. If we have a basis for the vector space of derivations on our ground field, then the problem of computing radicals can be reduced to computing p th roots of elements in finite dimensional algebras