Related papers: Dimension-Dependent Upper Bounds for Grobner Bases
In this note, we give a bound for the Castelnuovo-Mumford regularity of a homogeneous ideal $I$ in terms of the degrees of its generators. We assume that $I$ defines a local complete intersection with log canonical singularities.
Bounds on the Castelnuovo-Mumford regularity of the associated graded modules of k-Buchsbaum modules M are given in terms of k and some other invariants of M.
In the first chapter we present new results related on monomial ideals of Borel type. Also, we introduce a new class of monomial ideals, called $\de$-fixed ideals, which generalize the class of $p$-Borel ideals and we extend several results…
We establish uniform a-priori estimates for solutions of the semilinear Dirichlet problem \begin{equation} \begin{cases} (-\Delta)^m u=h(x,u)\quad&\mbox{in }\Omega,\\ u=\partial_nu=\cdots=\partial_n^{m-1}u=0\quad&\mbox{on }\partial\Omega,…
The purpose of this paper is to apply previous work on dormant opers to the study of the moduli space of stable bundles in positive characteristic. We affirmatively resolve the rank $2$ case of a conjecture proposed by the second author,…
In [4] Sturmfels linked the Hilbert Nullstellensatz to Gr\"obner bases through final polynomials. In (loc. cit.) it was claimed that final polynomials always appear in a lexicographic Gr\"obner basis of a certain ideal. In this paper, we…
We study existence and computability of finite bases for ideals of polynomials over infinitely many variables. In our setting, variables come from a countable logical structure A, and embeddings from A to A act on polynomials by renaming…
This is a revised version of the author's PhD thesis, including the corrections by the examiners. It also includes a few additional small corrections. In this thesis the objects of study are classifying spaces of groups with stabilisers in…
Matrix Schubert varieties are affine varieties arising in the Schubert calculus of the complete flag variety. We give a formula for the Castelnuovo-Mumford regularity of matrix Schubert varieties, answering a question of Jenna Rajchgot. We…
Let $S$ be a polynomial ring over any field $\Bbbk$, and let $P \subseteq S$ be a non-degenerate homogeneous prime ideal of height $h$. When $\Bbbk$ is algebraically closed, a classical result attributed to Castelnuovo establishes an upper…
In the article "Non-commutative Grobner bases for commutative algebras", Eisenbud-Peeva-Sturmfels proved a number of results regarding Grobner bases and initial ideals of those ideals in the free associative algebra which contain the…
We investigate the behavior of the higher-order degrees, db_n, of a finitely presented group G. These db_n are functions from H^1(G;Z) to Z whose values are the degrees certain higher-order Alexander polynomials. We show that if def(G) is…
The purpose of this paper is twofold. 1. We give combinatorial bounds on the ranks of the groups $\Tor^{R}_\bullet(k,k)_\bullet$ in the case where $R = k[\Lambda]$ is an affine semi-group ring, and in the process provide combinatorial…
We prove new upper bounds for the degrees in Hilbert's Nullstellensatz and for the Noether exponent of polynomial ideals in terms of the monomial structure of the polynomials involved. Our bounds improve the previously known bounds in the…
In this paper, we study the solving degrees for affine semi-regular sequences and their homogenized sequences. Some of our results are considered to give mathematically rigorous proofs of the correctness of methods for computing Gr\"{o}bner…
Suppose $I$ is an ideal of a polynomial ring over a field, $I\subseteq k[x_1,\ldots,x_n]$, and whenever $fg\in I$ with degree $\leq b$, then either $f\in I$ or $g\in I$. When $b$ is sufficiently large, it follows that $I$ is prime.…
Two fundamental questions in the theory of Groebner bases are decision ("Is a basis G of a polynomial ideal a Groebner basis?") and transformation ("If it is not, how do we transform it into a Groebner basis?") This paper considers the…
General revision. In particular the parts concerning involutive bases over rings have been significantly changed. In addition some proofs have been improved.
The $F$-thresholds are important numerical invariants in prime characteristic, whose existence had been established only under certain assumptions. We show the existence of $F$-thresholds in full generality. We study properties of standard…
Using symmetric algebras we simplify (and slightly strengthen) the Bruns-Eisenbud-Evans "generalized principal ideal theorem" on the height of order ideals of non-minimal generators in a module. We also obtain a simple proof and an…