Related papers: Matrix-F5 algorithms and tropical Gr\"obner bases …

Solving a system of $m$ multivariate quadratic equations in $n$ variables over finite fields (the MQ problem) is one of the important problems in the theory of computer science. The XL algorithm (XL for short) is a major approach for…

The tropicalization of a linear space over a non-archimedean field is a tropical linear space. In this paper, we present a method for computing the tropicalization of any lattice over a valuation ring. The resulting tropical semimodule is…

The tropical semiring is a semiring of extended real numbers, where the operations of `max' and `+' replace the usual addition and multiplication, respectively. Difference equations obtained from the ultradiscrete limit of discrete…

It has been shown previously that a large class of monomial maps equivariant under the action of an infinite symmetric group have finitely generated kernels up to the symmetric action. We prove that these symmetric toric ideals also have…

In this paper, a new rigorous numerical method to compute fundamental matrix solutions of non-autonomous linear differential equations with periodic coefficients is introduced. Decomposing the fundamental matrix solutions $\Phi(t)$ by their…

Floquet multipliers (characteristic multipliers) play significant role in the stability of the periodic equations. Based on the iterative method, we provide a unified algorithm to compute the Floquet multipliers (characteristic multipliers)…

Let p be an odd prime number and K be a p-adic field. In this paper, we develop an analogue of Fontaine's theory of (phi,Gamma)-modules replacing the p-cyclotomic extension by the extension K_infty obtained by adding to K a compatible…

Finding a common factor of two multivariate polynomials with approximate coefficients is a problem in symbolic-numeric computing. Taking a tropical view on this problem leads to efficient preprocessing techniques, applying polyhedral…

A generalization of recent group-theoretic matrix multiplication algorithms to an analogue of the theory of partial matrix multiplication is presented. We demonstrate that the added flexibility of this approach can in some cases improve…

Let T(x) in k[x] be a monic non-constant polynomial and write R=k[x] / (T) the quotient ring. Consider two bivariate polynomials a(x, y), b(x, y) in R[y]. In a first part, T = p^e is assumed to be the power of an irreducible polynomial p. A…

Given a zero-dimensional ideal I in K[x1,...,xn] of degree D, the transformation of the ordering of its Groebner basis from DRL to LEX is a key step in polynomial system solving and turns out to be the bottleneck of the whole solving…

Solving a polynomial system, or computing an associated Gr\"obner basis, has been a fundamental task in computational algebra. However, it is also known for its notorious doubly exponential time complexity in the number of variables in the…

We propose new algorithms for computing triangular decompositions of polynomial systems incrementally. With respect to previous works, our improvements are based on a {\em weakened} notion of a polynomial GCD modulo a regular chain, which…

Classically, Groebner bases are computed by first prescribing a set monomial order. Moss Sweedler suggested an alternative and developed a framework to perform such computations by using valuation rings in place of monomial orders. We build…

Let $\K$ be a field and $(f_1, \ldots, f_n)\subset \K[X_1, \ldots, X_n]$ be a sequence of quasi-homogeneous polynomials of respective weighted degrees $(d_1, \ldots, d_n)$ w.r.t a system of weights $(w_{1},\dots,w_{n})$. Such systems are…

In this article, we present a massively parallel framework for computing tropicalizations of algebraic varieties which can make use of finite symmetries. We compute the tropical Grassmannian TGr$_0(3,8)$, and show that it refines the…

Let F be the cyclotomic field of fifth roots of unity. We computationally investigate modularity of elliptic curves over F.

When we consider a finite abelian group acting linearly on a polynomial ring, we can find monomial generators for the subring of invariants. By Noether's degree bound and Hilbert's finiteness theorem, we know that there are finitely many…

We develop a Gr\"obner basis theory for a class of algebras that generalizes both PBW-algebras and rings of differential algebras on smooth varieties. Emphasis lies on methods to compute filtrations and graded structures defined by weight…

We develop a tropical analogue of the classical double description method allowing one to compute an internal representation (in terms of vertices) of a polyhedron defined externally (by inequalities). The heart of the tropical algorithm is…