We present a new algorithm to compute minimal telescopers for rational functions in two discrete variables. As with recent reduction-based approaches, our algorithm has the important feature that the computation of a telescoper is independent of its certificate. In addition, our algorithm uses a compact representation of the certificate, which allows it to be easily manipulated and analyzed without

In the area of symbolic-numerical computation within computer algebra, an interesting question is how “close” a random input is to the “critical” ones. Examples are the singular matrices in linear algebra or the polynomials with multiple roots for Newton's root-finding method. Bounds, sometimes very precise, are known for the volumes over R or C of such neighborhoods of the varieties of “critical”

Sparse polynomial interpolation, sparse linear system solving or modular rational reconstruction are fundamental problems in Computer Algebra. They come down to computing linear recurrence relations of a sequence with the Berlekamp–Massey algorithm. Likewise, sparse multivariate polynomial interpolation and multidimensional cyclic code decoding require guessing linear recurrence relations of a multivariate

We determine the probability, structure dependent, that the block Wiedemann algorithm correctly computes leading invariant factors. This leads to a tight lower bound for the probability, structure independent. We show, using block size slightly larger than r, that the leading r invariant factors are computed correctly with high probability over any field. Moreover, an algorithm is provided to compute

We describe a procedure for classifying positively graded filiform nilpotent Lie algebras in low dimensions, over R, C, and Q. Using this procedure, we find all filiform nilpotent Lie algebras of dimension 15 and lower that admit a positive grading.

We introduce tropical holonomic sequences of a given order and calculate their entropy in case of the second order.

We present a method to compute polynomial conservation laws for systems of partial differential equations (PDEs). The method only relies on linear algebraic computations and is complete, in the sense it can find a basis for all polynomial fluxes that yield conservation laws, up to a specified order of derivatives and degree. We compare our method to state-of-the-art algorithms based on the direct approach

The problem consists in bounding the number of tetrahedra that share the same volume V, circumradius R, and face areas A1,A2,A3,A4. A symbolic computation approach is used to prove that the upper bound of the number of tetrahedra is eight. First, the problem is reduced to a counting root problem using the Cayley-Menger determinant formula for the volume of a simplex and some metric equations of the

In 2013 Elkies described a method for bounding the transitivity degree of Galois groups. Our goal is to give additional applications of this technique, in particular verifying that the monodromy group of the degree-276 cover defined over a degree-12 number field computed by Monien is isomorphic to the sporadic Conway group Co3.

The groups whose orders factorise into at most four primes have been described (up to isomorphism) in various papers. Given such an order n, this paper exhibits a new explicit and compact determination of the isomorphism types of the groups of order n together with effective algorithms to enumerate, construct, and identify these groups. The algorithms are implemented for the computer algebra system

Let H≤Sn be an intransitive group with orbits Ω1,Ω2,…,Ωk. Then certainly H is a subdirect product of the direct product of its projections onto each orbit, H|Ω1×H|Ω2×…×H|Ωk. Here we provide a polynomial time algorithm for computing the finest partition P of the H-orbits such that H=∏c∈PH|c and we demonstrate its usefulness in some applications.

We show how the classical notions of cohomology with local coefficients, CW-complex, covering space, homeomorphism equivalence, simple homotopy equivalence, tubular neighbourhood, and spinning can be encoded on a computer and used to calculate ambient isotopy invariants of continuous embeddings N↪M of one topological manifold into another. More specifically, we describe an algorithm for computing the

In this paper, we prove a supercongruence via the Wilf–Zeilberger method and symbolic summation algorithms in the setting of difference rings. That is, for any prime p>3,∑n=0(p−1)/23n+1(−8)n(2nn)3≡p(−1p)+p34(2p)Ep−3(14)(modp4), where (⋅p) stands for the Legendre symbol, and En(x) are the Euler polynomials. This confirms a special case of a recent conjecture of Z.-W. Sun (Sun, 2019, (2.18)).

The sparse difference resultant introduced in Li et al. (2015b) is a basic concept in difference elimination theory. In this paper, we show that the sparse difference resultant of a generic Laurent transformally essential system can be computed via the sparse resultant of a simple algebraic system arising from the difference system. Moreover, new order bounds of sparse difference resultant are found

We consider the higher order Turán inequality and higher order log-concavity for sequences {an}n≥0 such thatan−1an+1an2=1+∑i=1mri(log⁡n)nαi+o(1nβ), where m is a nonnegative integer, αi are real numbers, ri(x) are rational functions of x and0<α1<α2<⋯<αm<β. We will give a sufficient condition on the higher order Turán inequality and the ℓ-log-concavity for n sufficiently large. Many P-recursive sequences

We consider the problem of computing exact sums of squares (SOS) decompositions for certain classes of non-negative multivariate polynomials, relying on semidefinite programming (SDP) solvers. We provide a hybrid numeric-symbolic algorithm computing exact rational SOS decompositions with rational coefficients for polynomials lying in the interior of the SOS cone. The first step of this algorithm computes

A new algorithm is introduced for computing logarithmic vector fields along a hypersurface with an isolated singularity. The key ideas of the algorithm are computing an ideal quotient in a polynomial ring and the use of algebraic local cohomology. The use of these ideas allows us to compute a module, over a local ring, of germs logarithmic vector fields. The resulting algorithms are much faster than

The Waring Problem over polynomial rings asks how to decompose a homogeneous polynomial p of degree d as a linear combination of d-th powers of linear forms. In this work we give an algorithm to obtain a real Waring decomposition of any given real binary form p of length at most its degree. In fact, we construct a semialgebraic family of Waring decompositions for p. We illustrate our results with some

As a fundamental notion, the free differential algebra on a set is concretely constructed as the polynomial algebra on the differential variables. Such a construction is not known for the more general notion of the free differential algebra on an algebra, from the left adjoint functor of the forgetful functor from differential algebras to algebras, instead of sets. In this paper we show that a generator-relation

We describe the qFunctions Mathematica package for q-series and partition theory applications. This package includes both experimental and symbolic tools. The experimental set of elements includes guessers for q-difference equations and recurrences for given q-difference and fitting/finding explicit expressions for sequences of polynomials. This package can symbolically handle formal manipulations

We present two new algorithms for the computation of the q-integer linear decomposition of a multivariate polynomial. Such a decomposition is essential for the treatment of q-hypergeometric symbolic summation via creative telescoping and for describing the q-counterpart of Ore-Sato theory. Both of our algorithms require only basic integer and polynomial arithmetic and work for any unique factorization

An ideal of polynomials is symmetric if it is closed under permutations of variables. We relate general symmetric ideals to the so called Specht ideals generated by all Specht polynomials of a given shape. We show a connection between the leading monomials of polynomials in the ideal and the Specht polynomials contained in the ideal. This provides applications in several contexts. Most notably, this

Vizing's conjecture (open since 1968) relates the product of the domination numbers of two graphs to the domination number of their Cartesian product graph. In this paper, we formulate Vizing's conjecture as a Positivstellensatz existence question. In particular, we select classes of graphs according to their number of vertices and their domination number and encode the conjecture as an ideal/polynomial

We present a complete algorithm that computes all hypergeometric solutions of homogeneous linear difference equations and rational solutions of parameterized linear difference equations in the setting of ΠΣ⁎-fields. More generally, we provide a flexible framework for a big class of difference fields that are built by a tower of ΠΣ⁎-field extensions over a difference field that enjoys certain algorithmic

Interpolation is a prime tool in algebraic computation while symmetry is a qualitative feature that can be more relevant to a mathematical model than the numerical accuracy of the parameters. The article shows how to exactly preserve symmetry in multivariate interpolation while exploiting it to alleviate the computational cost. We revisit minimal degree and least interpolation with symmetry adapted

In this paper we study matching in equational theories that specify counterparts of associativity and commutativity for variadic function symbols. We design a procedure to solve a system of matching equations and prove its termination, soundness, completeness, and minimality. The minimal complete set of matchers for such a system can be infinite, but our algorithm computes its finite representation

A symbolic Lie p-ring presentation defines a family of nilpotent Lie rings with pn elements for infinitely many primes p and a fixed positive integer n. Symbolic Lie p-ring presentations are used in the classification of isomorphism types of nilpotent Lie rings of order pn for all primes p and n≤7. We describe an algorithm to compute the Schur multipliers of all nilpotent Lie rings in the family defined

Feynman integral computations in theoretical high energy particle physics frequently involve square roots in the kinematic variables. Physicists often want to solve Feynman integrals in terms of multiple polylogarithms. One way to obtain a solution in terms of these functions is to rationalize all occurring square roots by a suitable variable change. In this paper, we give a rigorous definition of

We present a generic and executable formalization of signature-based algorithms (such as Faugère's F5) for computing Gröbner bases, as well as their mathematical background, in the Isabelle/HOL proof assistant. Said algorithms are currently the best known algorithms for computing Gröbner bases in terms of computational efficiency. The formal development attempts to be as generic as possible, generalizing

We introduce a “workable” notion of degree for non-homogeneous polynomial ideals and formulate and prove ideal theoretic Bézout inequalities for the sum of two ideals in terms of this notion of degree and the degree of generators.

We investigate the tree gonality of a genus-g metric graph, defined as the minimum degree of a tropical morphism from any tropical modification of the metric graph to a metric tree. We give a combinatorial constructive proof that this number is at most ⌈g/2⌉+1, a fact whose proofs so far required an algebro-geometric detour via special divisors on curves. For even genus, the tropical morphism which

It is well known that Riquier introduced the notion of multiplicative variables applied to order ideals to represent initial conditions of partial differential equations as series. On the other hands, Janet introduced the concept of involutive division. Following Riquier and Janet, we focus on the following problem. Suppose one needs to compute a monomial ideal generated in some degree D and its escalier

Modelling the substitution of nucleotides along a phylogenetic tree is usually done by a hidden Markov process. This allows to define a distribution of characters at the leaves of the trees and one might be able to obtain polynomial relationships among the probabilities of different characters. The study of these polynomials and the geometry of the algebraic varieties defined by them can be used to

We study Gröbner degenerations of Schubert varieties inside flag varieties. We consider toric degenerations of flag varieties induced by matching fields and semi-standard Young tableaux. We describe an analogue of matching field ideals for Schubert varieties inside the flag variety and give a complete characterization of toric ideals among them. We use a combinatorial approach to standard monomial

Both spectral sequences and persistent homology are tools in algebraic topology defined from filtrations of objects (e.g. topological spaces or simplicial complexes) indexed over the set Z of integer numbers. A recent work has shown the details of the relation between both concepts. Moreover, generalizations of both concepts have been proposed which originate from a different choice of the set of indices

Let K be a field of characteristic zero and let K‾ be an algebraic closure of K. Consider a sequence of polynomials G=(g1,…,gs) in K[X1,…,Xn] with s

Polynomial factoring has famous practical algorithms over fields– finite, rational and p-adic. However, modulo prime powers, factoring gets harder because there is non-unique factorization and a combinatorial blowup ensues. For example, x2+pmodp2 is irreducible, but x2+pxmodp2 has exponentially many factors in the input size (which here is logarithmic in p)! We present the first randomized poly(deg⁡f

We describe new fast algorithms for evaluation and interpolation on the “novel” polynomial basis over finite fields of characteristic two introduced by Lin et al. (2014). Fast algorithms are also described for converting between their basis and the monomial basis, as well as for converting to and from the Newton basis associated with the evaluation points of the evaluation and interpolation algorithms

An independence model for discrete random variables is a Segre-Veronese variety in a probability simplex. Any metric on the set of joint states of the random variables induces a Wasserstein metric on the probability simplex. The unit ball of this polyhedral norm is dual to the Lipschitz polytope. Given any data distribution, we seek to minimize its Wasserstein distance to a fixed independence model

We classify embedded blowups of the real affine plane up to oriented isomorphy. We show that two blowups in the same isomorphism class are isotopic, using a matrix deformation argument similar to an idea given in Shastri (2002). This answers two questions which were motivated by the interactive visualizations of such blowups (see Schenzel and Stussak, 2013, Stussak, 2007, 2013).

It is known since the 1970s that no more than 23 multiplications are required for computing the product of two 3×3-matrices. For non-commutative coefficient rings, it is not known whether it can also be done with fewer multiplications. However, there are several mutually inequivalent ways of doing the job with 23 multiplications. In this article, we extend this list considerably by providing more than

We classify the two-way quasi-independence models (independence models with structural zeros) that have rational maximum likelihood estimators, or MLEs. We give a necessary and sufficient condition on the bipartite graph associated to the model for the MLE to be rational. In this case, we give an explicit formula for the MLE in terms of combinatorial features of this graph. We also use the Horn uniformization

Given symmetric matrices A0,A1,…,An of size m with rational entries, the set of real vectors x=(x1,…,xn) such that the matrix A0+x1A1+⋯+xnAn has non-negative eigenvalues is called a spectrahedron. Minimization of linear functions over spectrahedra is called semidefinite programming. Such problems appear frequently in control theory and real algebra, especially in the context of nonnegativity certificates

The objective of this paper is twofold. First we do a survey on what we call the Yoshida method for studying the analytic first integrals of the quasi-homogeneous polynomial differential systems. After, we apply the Yoshida method for studying the analytic first integrals of all the quasi-homogeneous polynomial differential systems in R3 of degree 2.

Algorithms for integration of the algebraic functions implemented in modern computer algebra systems (CAS) are not always able to solve the classical problems of integration in the class of algebraic or elementary functions. The most general approach to describing the integral of an algebraic function is to find a standard representation for Abelian integrals, which, on the one hand, would not be too

In this paper, we study unirational differential curves and the corresponding differential rational parametrizations. We first investigate basic properties of proper differential rational parametrizations for unirational differential curves. Then we show that the implicitization problem of proper linear differential rational parametric equations can be solved by means of differential resultants. Furthermore

This paper introduces an algorithmic approach to the analysis of bifurcation of limit cycles from the centers of nonlinear continuous differential systems via the averaging method. We develop three algorithms to implement the averaging method. The first algorithm allows one to transform the considered differential systems to the normal form of averaging. Here, we restricted the unperturbed term of

We consider the problem of finding a condition for a univariate polynomial having a given multiplicity structure when the number of distinct roots is given. It is well known that such conditions can be written as conjunctions of several polynomial equations and one inequation in the coefficients, by using repeated parametric gcd's. In this paper, we give a novel condition which is not based on repeated

Zeilberger's method of creative telescoping is crucial for the computer-generated proofs of combinatorial and special-function identities. Telescopers are linear differential or (q-)recurrence operators computed by algorithms for creative telescoping. Two fundamental problems related to creative telescoping are whether telescopers exist, and how to construct them efficiently when they do. In this paper

We describe a new arithmetic system for the Magma computer algebra system for working with p-adic numbers exactly, in the sense that numbers are represented lazily to infinite p-adic precision. This is the first highly featured such implementation. This has the benefits of increasing user-friendliness and speeding up some computations, as well as forcibly producing provable results. We give theoretical

It is undecidable in general whether a given finitely presented group is word hyperbolic. We use the concept of pregroups, introduced by Stallings (1971), to define a new class of van Kampen diagrams, which represent groups as quotients of virtually free groups. We then present a polynomial-time procedure that analyses these diagrams, and either returns an explicit linear Dehn function for the presentation

Square-free decomposition is one of fundamental computations for polynomials. However, any conventional algorithm may not work for polynomials with a priori errors on their coefficients. There are mainly two approaches to overcome this empirical situation: approximate polynomial GCD (greatest common divisor) and the nearest singular polynomial. In this paper, we show that these known approaches are

We provide algorithms to reconstruct rational ruled surfaces in three-dimensional projective space from the “apparent contour” of a single projection to the projective plane. We deal with the case of tangent developables and of general projections to P3 of rational normal scrolls. In the first case, we use the fact that every such surface is the projection of the tangent developable of a rational normal

Multiplicative relations between the roots of a polynomial in Q[x] have drawn much attention in the field of arithmetic and algebra, while the problem of computing these relations is interesting to researchers in many other fields. In this paper, a sufficient condition is given for a polynomial f∈Q[x] to have only trivial multiplicative relations between its roots, which is a generalization of those

Let p be an odd natural number ≥3. Inspired by results from Euclid's Elements, we express the irrationaly=d+Rp, whose degree is 2p, as a polynomial function of irrationals of degrees ≤p. In certain cases y is expressed by simple radicals. This reduction of the degree exhibits remarkably regular patterns of the polynomials involved. The proof is based on hypergeometric summation, in particular, on Zeilberger's

We introduce the notion of confinement of decompositions for forms or vector of forms. The confinement, when it holds, lowers the number of parameters that one needs to consider, in order to find all the possible decompositions of a given set of data. With the technique of confinement, we obtain here two results. First, we give a new, shorter proof of a result by London (1890) that 3 general plane

We present a new formula for the Cauchy index of a rational function on an interval using subresultant polynomials. There is no condition on the endpoints of the interval and the formula also involves in some cases less subresultant polynomials.

For a given projective variety X, the high rank loci are the closures of the sets of points whose X-rank is higher than the generic one. We show examples of strict inclusion arising from two consecutive high rank loci. Our first example comes from looking at the Veronese surface of plane quartics. Although Piene had already shown an example in which X is a curve, we construct infinitely many curves

Consider an action of a connected solvable group G on an affine variety X. This paper presents an algorithm that constructs a semi-invariant f∈K[X]=:R and computes the invariant ring (Rf)G together with a presentation. The morphism Xf→Spec((Rf)G) obtained from the algorithm is a universal geometric quotient. In fact, it is even better than that: a so-called excellent quotient. If R is a polynomial

Every real algebraic variety determines a Voronoi decomposition of its ambient Euclidean space. Each Voronoi cell is a convex semialgebraic set in the normal space of the variety at a point. We compute the algebraic boundaries of these Voronoi cells.