Related papers: Separating Variables in Bivariate Polynomial Ideal…

For a given ideal I in K[x_1,...,x_n,y_1,...,y_m] in a polynomial ring with n+m variables, we want to find all elements that can be written as f-g for some f in K[x_1,...,x_n] and some g in K[y_1,...,y_m], i.e., all elements of I that…

We present a semi-algorithm which for any irreducible $p\in\mathbb{K}[x,y]$ finds all elements of $\mathbb{K}(x) + \mathbb{K}(y)$ that are of the form $qp$ for some $q\in\mathbb{K}(x,y)$ whose denominator is not divisible by $p$.

We present an algorithm to decide whether a given ideal in the polynomial ring contains a monomial without using Gr\"obner bases, factorization or sub-resultant computations.

Proving statements about linear operators expressed in terms of identities often leads to finding elements of certain form in noncommutative polynomial ideals. We illustrate this by examples coming from actual operator statements and…

Given a $d \times n$ integer matrix $A$, the main result is an elementary, simple-to-state algorithm that finds the largest $A$-graded ideal contained in any ideal $I$ in a polynomial ring $\Bbbk[x_1,\ldots,x_n]$. The special case where $A$…

The reduction number of monomial ideals in the polynomial $K[x,y]$ is studied. We focus on ideals $I$ for which $J=(x^a,y^b)$ is a reduction ideal. The computation of the reduction number amounts to solve linear inequalities. In some…

Binomial ideals are special polynomial ideals with many algorithmically and theoretically nice properties. We discuss the problem of deciding if a given polynomial ideal is binomial. While the methods are general, our main motivation and…

We will study monomial ideals $I$ in the exterior algebra as well as in the polynomial ring whose generic initial ideal is constant for all term orders up to permutations of variables. First, in the exterior algebra, we determine all graphs…

Let $\mathcal{R} = \mathbb{K}[x_1, \dots, x_n]$ be a multivariate polynomial ring over a field $\mathbb{K}$ of characteristic 0. Consider $n$ algebraically independent elements $g_1, \dots, g_n$ in $\mathcal{R}$. Let $\mathcal{S}$ denote…

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…

We investigate the structure of ideals generated by binomials (polynomials with at most two terms) and the schemes and varieties associated to them. The class of binomial ideals contains many classical examples from algebraic geometry, and…

It is well known that the multiplier ideal $\multr{I}$ of an ideal $I$ determines in a straightforward way the multiplier ideal $\multr{f}$ of a sufficiently general element $f$ of $I$. We give an explicit condition on a polynomial $f \in…

Let $K$ be the number field determined by a monic irreducible polynomial $f(x)$ with integer coefficients. In previous papers we parameterized the prime ideals of $K$ in terms of certain invariants attached to Newton polygons of higher…

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…

A new algorithm is presented for computing the largest degree invariant factor of the Sylvester matrix (with respect either to $x$ or $y$) associated to two polynomials $a$ and $b$ in $\mathbb F_q[x,y]$ which have no non-trivial common…

In this note, we prove that if $A$ is a finite set of real numbers such that $|AA| = K|A|$, then for every polynomial $f \in \mathbb{R}[x,y]$ we have that $|f(A,A)| = \Omega_{K,\operatorname{deg} f}(|A|^2)$, unless $f$ is of the form…

This paper investigates atomic factorizations in the monoid $\mathcal I(R)$ of nonzero ideals of a multivariate polynomial ring $R$, under ideal multiplication. Building on recent advances in factorization theory for unit-cancellative…

Let $R=K[x_1,\ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I$ be a polymatroidal ideal of $R$. In this paper, we provide a comprehensive classification of all unmixed polymatroidal ideals. This work…

If $I$ is a monomial ideal with linear quotients, then it has componentwise linear quotients. However, the converse of this statement is an open question. In this paper, we provide two classes of ideals for which the converse of this…

We provide an algorithm that computes a set of generators for any complete ideal in a smooth complex surface. More interestingly, these generators admit a presentation as monomials in a set of maximal contact elements associated to the…