Related papers: Computing and Using Minimal Polynomials

In this article, we study the primary decomposition of some binomial ideals. In particular, we introduce the concept of polyocollection, a combinatorial object that generalizes the definitions of collection of cells and polyomino, that can…

Resultants and Gr\"obner bases are crucial tools in studying polynomial elimination theory. We investigate relations between the variety of the resultant of two polynomials and the variety of the ideal they generate. Then we focus on the…

We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. Normal form algorithms provide an algebraic approach to solve this problem.…

Let $f(Z)=Z^n-a_{1}Z^{n-1}+\cdots+(-1)^{n-1}a_{n-1}Z+(-1)^na_n$ be a monic polynomial with coefficients in a ring~$R$ with identity, not necessarily commutative. We study the ideal $I_f$ of $R[X_1,\dots,X_n]$ generated by…

Let $k$ be an infinite field and $I\subset k [x_1, \ldots ,x_n]$ be an ideal such that dim $V(I)=q$. Denote by $(f_1, \ldots, f_s)$ a set of generators of $I$. One can see that in the set $I\cap k [x_{1},...,x_{q+1}]$ there exist non-zero…

We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. We propose a general algebraic framework to find the solutions and to…

In this paper, we focus on computing local minimizers of a multivariate polynomial optimization problem under certain genericity conditions. By using a technique in computer algebra and the second-order optimality condition, we provide a…

We determine, in a polynomial ring over a field, the arithmetical rank of certain ideals generated by a set of monomials and one binomial.

Let $n\in\mathbb{N}$ be fixed, $Q>1$ be a real parameter and $\mathcal{P}_n(Q)$ denote the set of polynomials over $\mathbb{Z}$ of degree $n$ and height at most $Q$. In this paper we investigate the following counting problems regarding…

The problem of evaluating potential integrals on planar triangular elements has been addressed using a polar coordinate decomposition. The resulting formulae are general, exact, easily implemented, and have only one special case, that of a…

In this paper, we are concerned with the problem of counting the multiplicities of a zero-dimensional regular set's zeros. We generalize the squarefree decomposition of univariate polynomials to the so-called pseudo squarefree decomposition…

This paper studies the hierarchy of local minimums of a polynomial in the space. For this purpose, we first compute H-minimums, for which the first and second order optimality conditions are satisfied. To compute each H-minimum, we…

We present a new probabilistic algorithm to compute modular polynomials modulo a prime. Modular polynomials parameterize pairs of isogenous elliptic curves and are useful in many aspects of computational number theory and cryptography. Our…

The core of an ideal is the intersection of all its reductions. We describe the core of a zero-dimensional monomial ideal I as the largest monomial ideal contained in a general reduction of I. This provides a new interpretation of the core…

For an arbitrary ideal $I$ in a polynomial ring $R$ we define the notion of initially regular sequences on $R/I$. These sequences share properties with regular sequences. In particular, the length of an initially regular sequence provides a…

Multi-homogeneous polynomial systems arise in many applications. We provide bit complexity estimates for solving them which, up to a few extra other factors, are quadratic in the number of solutions and linear in the height of the input…

Let $S=\mathbb{K}[x_1,\ldots,x_n]$ the polynomial ring over a field $\mathbb{K}$. In this paper for some families of monomial ideals $I \subset S$ we study the minimal number of generators of $I^k$. We use this results to find some other…

We investigate the computational complexity of deciding whether a given univariate integer polynomial p(x) has a factor q(x) satisfying specific additional constraints. When the only constraint imposed on q(x) is to have a degree smaller…

Given a polynomial ring $C$ over a field and proper ideals $I$ and $J$ whose generating sets involve disjoint variables, we determine how to embed the associated primes of each power of $I+J$ into a collection of primes described in terms…

A new efficient algorithm is proposed for factoring polynomials over an algebraic extension field. The extension field is defined by a polynomial ring modulo a maximal ideal. If the maximal ideal is given by its Groebner basis, no extra…