Related papers: Algorithms for Checking Zero-Dimensional Complete …

Wiebe's criterion, which recognizes complete intersections of dimension zero among the class of noetherian local rings, is revisited and exploited in order to provide information on what we call C.I.0-ideals (those such that the…

Given a set $\mathcal A = \{a_1,\ldots,a_n\} \subset \mathbb{N}^m$ of nonzero vectors defining a simplicial toric ideal $I_{\mathcal A} \subset k[x_1,...,x_n]$, where $k$ is an arbitrary field, we provide an algorithm for checking whether…

This paper studies algebraic residual intersections in rings with Serre's condition \( S_{s} \). It demonstrates that residual intersections admit free approaches i.e. perfect subideal with the same radical. This fact leads to determining a…

Throughout this abstruct $A$ will denote a noetherian commutative ring of dimension $n$. The paper has two parts. Among the interesting results in Part-1 are the following: 1) {\it suppose that $f_1, f_2, ..., f_r$ (with $r \leq n$) is a…

We study the complete intersection property and the algebraic invariants (index of regularity, degree) of vanishing ideals on degenerate tori over finite fields. We establish a correspondence between vanishing ideals and toric ideals…

We consider complete intersection ideals in a polynomial ring over a field of characteristic zero that are stable under the action of the symmetric group permuting the variables. We determine the possible representation types for these…

A good way of parametrizing 0-dimensional schemes in an affine space $\mathbb{A}_K^n$ has been developed in the last 20 years using border basis schemes. Given a multiplicity $\mu$, they provide an open covering of the Hilbert scheme ${\rm…

An almost complete intersection ideal can be seen as a $d$-sequence ideal with the minimal number of generators being one more than its height. In this paper, we give exact formulas for the regularity of powers of graded almost complete…

Border basis schemes are open subschemes of Hilbert schemes parametrizing 0-dimensional subschemes of $\mathbb{P}^n$ of given length. They yield open coverings and are easy to describe and to compute with. Our topic is to find re-embeddings…

$V$ is a complete intersection scheme in a multiprojective space if it can be defined by an ideal $I$ with as many generators as $\textrm{codim}(V)$. We investigate the multigraded regularity of complete intersections scheme in…

A commutative local ring is generally defined to be a complete intersection if its completion is isomorphic to the quotient of a regular local ring by an ideal generated by a regular sequence. It has not previously been determined whether…

Let $k$ be a commutative ring and $S=k[x_0, \ldots, x_n]$ be a polynomial ring over $k$ with a monomial order. For any monomial ideal $J$, there exists an affine $k$-scheme of finite type, called Gr\"obner scheme, which parameterizes all…

Given an affine algebra $R=P/I$, where $P=K[x_1,\dots,x_n]$ is a polynomial ring over a field $K$ and $I$ is an ideal in $P$, we study re-embeddings of the affine scheme ${\rm Spec}(R)$, i.e., presentations $R \cong P'/I'$ such that $P'$ is…

Let R be monomial sub-algebra of $k[x_1,...,x_N]$ generated by square free monomials of degree two. This paper addresses the following question: when is R a complete intersection? For such a k-algebra we can associate a graph G whose…

For zero-dimensional complete intersections with homogeneous ideal generators of equal degrees over an algebraically closed field of characteristic zero, we give a combinatorial proof of the smoothness of the corresponding catalecticant…

We prove that every local complete intersection curve in $Spec(A)$, where $A$ is a commutative Noetherian ring of dimension three, is a set-theoretic complete intersection. An analogous result is established for local complete intersection…

Given an affine algebra $R=K[x_1,\dots,x_n]/I$ over a field $K$, where $I$ is an ideal in the polynomial ring $P=K[x_1,\dots,x_n]$, we examine the task of effectively calculating re-embeddings of $I$, i.e., of presentations $R=P'/I'$ such…

For an ideal $I\subseteq\mathbb{R}[x]$ given by a set of generators, a new semidefinite characterization of its real radical $I(V_\mathbb{R}(I))$ is presented, provided it is zero-dimensional (even if $I$ is not). Moreover we propose an…

The Hilbert polynomial of a homogeneous complete intersection is determined by the degrees of the generators of the defining ideal. The degrees of the generators are not, in general, determined by the Hilbert polynomial -- but sometimes…

Consider a complete intersection I of type (d_1,..., d_r) in a polynomial ring over a field of characteristic 0. We study the graded system of ideals {gin(I^n)}_n obtained by taking the reverse lexicographic generic initial ideals of the…