Related papers: Numerical Algorithms for Dual Bases of Positive-Di…
An ideal $I$ in a Noetherian ring is called \textit{normal} if $I^n$ is integrally closed for all $n \geq 1$. Zariski proved that in two-dimensional regular local rings, every integrally closed ideal is normal. However, in dimension three…
A chopped ideal is obtained from a homogeneous ideal by considering only the generators of a fixed degree. We investigate cases in which the chopped ideal defines the same finite set of points as the original one-dimensional ideal. The…
The number of equations needed to cut out a variety given by an ideal is called the arithmetic rank (of the ideal). It was shown in [8] that the notion of arithmetic rank is strongly related to the concept of regular sequences on the Matlis…
We determine, in a polynomial ring over a field, the arithmetical rank of certain ideals generated by a set of monomials and one binomial.
Given a symbolic power of a homogeneous ideal in a polynomial ring, we study the problem of determining which powers of the ideal contain it. For ideals defining 0-dimensional subschemes of projective space, as an immediate corollary of our…
In this paper we describe an efficient involutive algorithm for constructing Groebner bases of polynomial ideals. The algorithm is based on the concept of involutive monomial division which restricts the conventional division in a certain…
This paper is concerned with linear algebra based methods for solving exactly polynomial systems through so-called Gr\"obner bases, which allow one to compute modulo the polynomial ideal generated by the input equations. This is a topical…
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.
It is known that a linear code can be represented by a binomial ideal. In this paper, we give standard bases for the ideals in a localization of the multivariate polynomial ring in the case of linear codes over prime fields.
We consider the problem of mutually unbiased bases as a polynomial optimization problem over the reals. We heavily reduce it using known symmetries before exploring it using two methods, combining a number of optimization techniques. The…
Given a finite set of arbitrarily distributed points in affine space with arbitrary multiplicity structures, we present an algorithm to compute the reduced Groebner basis of the vanishing ideal under the lexicographic ordering. Our method…
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…
Dimensional regularization is applied to the Lippmann-Schwinger equation for a separable potential which gives rise to logarithmic singularities in the Born series. For this potential a subtraction at a fixed energy can be used to…
Let $R^h$ denote the polynomial ring in variables $x_1,\,\ldots,\, x_h$ over a specified field $K$. We consider all of these rings simultaneously, and in each use lexicographic (lex) monomial order with $x_1 > \cdots > x_h$. Given a fixed…
In this article we give an explicit formula for the jumping numbers of an ideal of finite colenght in a two-dimensional regular local ring with an algebraically closed residue field. For this purpose, we associate a certain numerical…
The properties of the intersection algebra of two principal monomial ideals in a polynomial ring are investigated in detail. Results are obtained regarding the Hilbert series and the canonical ideal of the intersection algebra using methods…
In this article, we study the jumping numbers of an ideal in the local ring at rational singularity on a complex algebraic surface. By understanding the contributions of reduced divisors on a fixed resolution, we are able to present an…
Hilbert schemes of zero-dimensional ideals in a polynomial ring can be covered with suitable affine open subschemes whose construction is achieved using border bases. Moreover, border bases have proved to be an excellent tool for describing…
We present a new algorithm for computing the real radical of an ideal and, more generally, the-radical of, which is based on convex moment optimization. A truncated positive generic linear functional vanishing on the generators of is…
In classical and real algebraic geometry there are several notions of the radical of an ideal I. There is the vanishing radical defined as the set of all real polynomials vanishing on the real zero set of I, and the real radical defined as…