Invariants withstand transformations and, therefore, represent the essence of objects or phenomena. In mathematics, transformations often constitute a group action. Since the 19th century, studying the structure of various types of invariants and ...

Hypergeometric sequences are rational-valued sequences that satisfy first-order linear recurrence relations with polynomial coefficients; that is, a hypergeometric sequence is one that satisfies a recurrence of the form f(n)un = g(n)un − 1 where .

In ...

We introduce a new method for discovering matrix multiplication schemes based on random walks in a certain graph, which we call the flip graph. Using this method, we were able to reduce the number of multiplications for the matrix formats (4,4,5) and (5,...

A “blendstring” is a piecewise polynomial interpolant with high-degree two-point Hermite interpolational polynomials on each piece, analogous to a cubic spline. Blendstrings are smoother and can be more accurate than cubic splines, and can be used to ...

For computing the greatest common divisor (GCD) of a given set of multivariate polynomials, in general we use one of modular algorithms to avoid any growth in the coefficient polynomials in the intermediate expressions. However, for computing ...

Finding roots of a bivariate polynomial f(x1, x2), over a prime field , is a fundamental question with a long history and several practical algorithms are now known. Effective algorithms for describing the roots modulo pk, k ≥ 2, for any general ...

In 2013, Pak and Panova proved the strict unimodality property of q-binomial coefficients (as polynomials in q) based on the combinatorics of Young tableaux and the semigroup property of Kronecker coefficients. They showed it to be true for all ℓ, m ≥ 8 ...

A universal analytic Gröbner basis (UAGB) of an ideal of a Tate algebra is a set containing a local Gröbner basis for all suitable convergence radii. In a previous article, the authors proved the existence of finite UAGB’s for polynomial ideals, leaving ...

A decision algorithm for the first-order theory of univariate mixed trigonometric-polynomials over the reals is proposed in this paper. In the development of the decision algorithm, the concept "contraction mapping associated with an algebraic function" ...

The Empty Integral Hull problem is a feasibility problem that is a linear representation of a subset of the Boolean Hypercube with n vertices with no integral points. In this paper, we study the Sum of Squares rank of the problem. We present a novel ...

Given a system of analytic functions and an approximation to a cluster of zeros, we wish to construct two regions containing the cluster and no other zeros of the system. The smaller region tightly contains the cluster while the larger region separates ...

We investigate geometric embeddings among several classes of generalised cone complexes and algorithms, e.g., to compute their homology. Interesting cases arise from moduli spaces of tropical curves. Specifically, via an explicit computation, we show ...

Trager’s Hermite reduction solves the integration problem for algebraic functions via integral bases. A generalization of this algorithm to D-finite functions has so far been limited to the Fuchsian case. In the present paper, we remove this restriction ...

Although in theory we can decide whether a given D-finite function is transcendental, transcendence proofs remain a challenge in practice. Typically, transcendence is certified by checking certain incomplete sufficient conditions. In this paper we ...

McCallum-style Cylindrical Algebra Decomposition (CAD) is a major improvement on the original Collins version, and has had many subsequent advances, notably for total or partial equational constraints. But it suffers from a problem with nullification. ...

In this paper, we present an algorithm that enumerates a certain class of signed permutations, referred to as grid signed permutation classes. In the case of permutations, the corresponding grid classes are of interest because they are equivalent to the ...

We present a p-adic algorithm to recover the lexicographic Gröbner basis of an ideal in with a generating set in , with a complexity that is less than cubic in terms of the dimension of and softly linear in the height of its coefficients. We observe ...

This paper continues the studies of symbolic integration by focusing on the stability problems on D-finite functions. We introduce the notion of stability index in order to investigate the order growth of the differential operators satisfied by iterated ...

We present a new algorithm for computing the characteristic polynomial of an arbitrary endomorphism of a finite Drinfeld module using its associated crystalline cohomology. Our approach takes inspiration from Kedlaya’s p-adic algorithm for computing the ...

The usual formulation of efficient division uses Newton iteration to compute an inverse in a related domain where multiplicative inverses exist. On one hand, Newton iteration allows quotients to be calculated using an efficient multiplication method. On ...