Related papers: Tame graphs, clutters and their Rees algebras

We study the group Tame($\mathbf A^3$) of tame automorphisms of the 3-dimensional affine space, over a field of characteristic zero. We recover, in a unified and (hopefully) simplified way, previous results of Kuroda, Shestakov, Umirbaev…

Symmetric strongly shifted ideals are a class of monomial ideals which come equipped with an action of the symmetric group and are analogous to the well-studied class of strongly stable monomial ideals. In this paper we focus on algebraic…

In this paper, we introduce the notion of binomial edge ideals of a clutter and obtain results similar to those obtained for graphs by Rauf \& Rinaldo in \cite{raufrin}. We also answer a question posed in their paper.

An ideal $I$ of a commutative ring $R$ is said to be of linear type when its Rees algebra and symmetric algebra exhibit isomorphism. In this paper, we investigate the conjecture put forth by Jayanthan, Kumar, and Sarkar (2021) that if $G$…

Well ordered covers of square-free monomial ideals are subsets of the minimal generating set ordered in a certain way that give rise to a Lyubeznik resolution for the ideal, and have guaranteed nonvanishing Betti numbers in certain degrees.…

We study ideals which are generated by monomials of degree $d$ in the polynomial ring in $n$ variables and which satisfy certain numerical side conditions regarding their exponents. Typical examples of such ideals are the ideals of Veronese…

The aim of this thesis is to investigate the Betti diagrams of squarefree monomial ideals in polynomial rings. We use two key tools to help us study these diagrams. The first is the Stanley-Reisner Correspondence, which assigns a unique…

Let $a_{i1}x_1+\cdots+a_{ik}x_k=0$, $i\in[m]$ be a balanced homogeneous system of linear equations with coefficients $a_{ij}$ from a finite field $\mathbb{F}_q$. We say that a solution $x=(x_1,\ldots, x_k)$ with $x_1,\ldots, x_k\in…

We study when Taylor resolutions of monomial ideals are minimal. We consider monomial ideals with linear quotients. In particular, we determine precisely the stable ideals and the monomial ideals with linear resolutions having the miminal…

Let G be a perfect graph and let J be its ideal of vertex covers. We show that the Rees algebra of J is normal and that this algebra is Gorenstein if G is unmixed. Then we give a description--in terms of cliques--of the symbolic Rees…

We construct the first linear strand of the minimal free resolutions of edge ideals of $d$-partite $d$-uniform clutters. We show that the first linear strand is supported on a relative simplicial complex. In the case that the edge ideals of…

Let I be a homogeneous ideal of a polynomial ring S. We prove that if the initial ideal J of I, w.r.t. a term order on S, is square-free, then the extremal Betti numbers of S/I and of S/J coincide. In particular, depth(S/I)=depth(S/J) and…

We study the regularity of symbolic powers of square-free monomial ideals. We prove that if $I = I_\Delta$ is the Stanley-Reisner ideal of a simplicial complex $\Delta$, then $\reg(I^{(n)}) \leqslant \delta(n-1) +b$ for all $n\geqslant 1$,…

A univariate polynomial f over a field is decomposable if f = g o h = g(h) for nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials over a finite field. The…

Normal ideals on regular uncountable cardinals are familiar objects. We investigate ideals that are pleasant--while a normal ideal is closed under arbitrary diagonal unions, a pleasant ideal is closed only under diagonal unions indexed by…

Given two finite sequences of positive integers $\alpha$ and $\beta$, we associate a square free monomial ideal $I_{\alpha,\beta}$ in a ring of polynomials $S$, and we recursively compute the algebraic invariants of $S/I_{\alpha,\beta}$.…

Let $S={\sf k}[X_1,\dots, X_n]$ be a polynomial ring, where ${\sf k}$ is a field. This article deals with the defining ideal of the Rees algebra of squarefree monomial ideal generated in degree $n-2$. As a consequence, we prove that Betti…

We introduce the notion of sortability and $t$-sortability for a simplicial complex and study the graphs for which their independence complexes are either sortable or $t$-sortable. We show that the proper interval graphs are precisely the…

We give several versions of local and global inverse mapping theorem for tame non necessarily smooth, mappings. Here tame mapping means a mapping which is subanalytic or, more generally, definable in some o-minimal structure. Our sufficient…

We show that a tilted algebra $A$ is tame if and only if for each generic root $\dd$ of $A$ and each indecomposable irreducible component $C$ of $\module(A,\dd)$, the field of rational invariants $k(C)^{\GL(\dd)}$ is isomorphic to $k$ or…