Related papers: Computing syzygies in finite dimension using fast …

Let $(G_n(x))_{n=0}^\infty$ be a $d$-th order linear recurrence sequence having polynomial characteristic roots, one of which has degree strictly greater than the others. Moreover, let $m\geq 2$ be a given integer. We ask for…

We consider $m \times s$ matrices (with $m\geq s$) in a real affine subspace of dimension $n$. The problem of finding elements of low rank in such spaces finds many applications in information and systems theory, where low rank is…

This article deals with the computation of the characteristic polynomial of dense matrices over small finite fields and over the integers. We first present two algorithms for the finite fields: one is based on Krylov iterates and Gaussian…

We describe a provably quasi-polynomial algorithm to compute discrete logarithms in the multiplicative groups of finite fields of small characteristic, that is finite fields whose characteristic is logarithmic in the order. We partially…

In this paper, we construct several new permutation polynomials over finite fields. First, using the linearized polynomials, we construct the permutation polynomial of the form $\sum_{i=1}^k(L_{i}(x)+\gamma_i)h_i(B(x))$ over ${\bf…

We consider two kinds of problems: the computation of polynomial and rational solutions of linear recurrences with coefficients that are polynomials with integer coefficients; indefinite and definite summation of sequences that are…

Let K be a subfield of the complex numbers, and let D be the Weyl algebra of K-linear differential operators on K[x_1,...,x_n]. If M and N are holonomic left D-modules we present an algorithm that computes explicit generators for the finite…

In this paper we consider systems of partial (multidimensional) linear difference equations. Specifically, such systems arise in scientific computing under discretization of linear partial differential equations and in computational high…

Signature-based algorithms have become a standard approach for computing Gr\"obner bases in commutative polynomial rings. However, so far, it was not clear how to extend this concept to the setting of noncommutative polynomials in the free…

In this paper, we achieve three primary objectives related to the rigorous computational analysis of nonlinear PDEs posed on complex geometries such as disks and cylinders. First, we introduce a validated Matrix Multiplication Transform…

We study the complexity of Gr\"obner bases computation, in particular in the generic situation where the variables are in simultaneous Noether position with respect to the system. We give a bound on the number of polynomials of degree $d$…

We give criteria for finite dimensionality or infinite dimensionality of the polynomial centralizer of the Lie algebra of a linear Lie group, in terms of invariants and relative invariants of the group. In the finite dimensional scenario…

Suppose $F$ is an infinite field and let $f \in F\{X_1, \dots,X_m\}$ be a noncommutative polynomial. Partially answering a query of Makar-Limanov, we show that there are numbers $d$ and $m'$ such that, if $F$ is closed under taking $d$th…

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…

Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that,…

Montgomery modular multiplication is widely-used in public key cryptosystems (PKC) and affects the efficiency of upper systems directly. However, modulus is getting larger due to the increasing demand of security, which results in a heavy…

We give a new complexity bound for calculating the complex dimension of an algebraic set. Our algorithm is completely deterministic and approaches the best recent randomized complexity bounds. We also present some new, significantly sharper…

If $A$ is a finite-dimensional algebra graded by a group $G$, and $\sigma \in G$, we define a variant of paratrophic matrix associated with $A$ and $\sigma$, and we use it to characterize the $\sigma$-graded Frobenius property for $A$. We…

We propose a general algorithm of constructing an extended formulation for any given set of linear constraints with integer coefficients. Our algorithm consists of two phases: first construct a decision diagram $(V,E)$ that somehow…

Given a compact parameter set $Y\subset R^p$, we consider polynomial optimization problems $(P_y$) on $R^n$ whose description depends on the parameter $y\inY$. We assume that one can compute all moments of some probability measure $\phi$ on…