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The need to know a few singular triplets associated with the largest singular values of third-order tensors arises in data compression and extraction. This paper describes a new method for their computation using the t-product. Methods for…
Here the polynomial interpolation approach is used to introduce the main results on multivariate normal algebraic systems. Next we bring a construction which shows that any standard algebraic system, with finite set of solutions, can be…
The Waring Problem over polynomial rings asks for how to decompose an homogeneous polynomial of degree $d$ as a finite sum of $d^{th}$ powers of linear forms. First, we give a constructive method to obtain a real Waring decomposition of any…
We characterize the fixed divisor of a polynomial $f(X)$ in $\mathbb{Z}[X]$ by looking at the contraction of the powers of the maximal ideals of the overring ${\rm Int}(\mathbb{Z})$ containing $f(X)$. Given a prime $p$ and a positive…
Counting Euclidean triangulations with vertices in a finite set $\C$ of the convex hull $\conv(\C)$ of $\C$ is difficult in general, both algorithmically and theoretically. The aim of this paper is to describe nearly convex polygons, a…
A computation method of algebraic local cohomology with parameters, associated with zero-dimensional ideal with parameter, is introduced. This computation method gives us in particular a decomposition of the parameter space depending on the…
An ideal is a classical object of study in the field of algebraic number theory. In maximal quadratic orders of number fields, ideals usually represented by the $\mathbb Z$-basis. This form of representation is used in most of the…
We introduce a new, quadratically convergent algorithm for finding maximum absolute value entries of tensors represented in the canonical format. The computational complexity of the algorithm is linear in the dimension of the tensor. We…
We describe an algorithm which finds binomials in a given ideal $I\subset\mathbb{Q}[x_1,\dots,x_n]$ and in particular decides whether binomials exist in $I$ at all. Binomials in polynomial ideals can be well hidden. For example, the lowest…
We consider the problem of finding exact sums of squares (SOS) decompositions for certain classes of non-negative multivariate polynomials, relying on semidefinite programming (SDP) solvers. We start by providing a hybrid numeric-symbolic…
Let $\mathcal{A}$ be an affine hyperplane arrangement, $L(\mathcal{A})$ its intersection poset, and $\chi_{\mathcal{A}}(t)$ its characteristic polynomial. This paper aims to propose combinatorial structures for the factorization of…
A reformulation of the three circles theorem of Johnson with distance coordinates to the vertices of a triangle is explicitly represented in a polynomial system and solved by symbolic computation. A similar polynomial system in distance…
Let $R=k[x,y,z]$ be a standard graded $3$-variable polynomial ring, where $k$ denotes any field. We study grade $3$ homogeneous ideals $I \subseteq R$ defining compressed rings with socle $k(-s)^{\ell} \oplus k(-2s+1)$, where $s \geq3$ and…
In this paper, we consider the problem of representing a multivariate polynomial as the determinant of a definite (monic) symmetric/Hermitian linear matrix polynomial (LMP). Such a polynomial is known as determinantal polynomial.…
An effective method to obtain exact analytical solutions of equations describing the coherent dynamics of multilevel systems is presented. The method is based on the usage of orthogonal polynomials, integral transforms and their discrete…
We discuss the problem of determining reduction number of a polynomial ideal I in n variables. We present two algorithms based on parametric computations. The first one determines the absolute reduction number of I and requires computation…
We give bit-size estimates for the coefficients appearing in triangular sets describing positive-dimensional algebraic sets defined over Q. These estimates are worst case upper bounds; they depend only on the degree and height of the…
Computing the determinant of a matrix with the univariate and multivariate polynomial entries arises frequently in the scientific computing and engineering fields. In this paper, an effective algorithm is presented for computing the…
A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors of a given rank. By moving away from computing polynomials via elimination theory to computing pseudowitness…
Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral…