arXiv subjects

Yairon Cid-Ruiz

Publications and source records attributed to Yairon Cid-Ruiz.

41 records · Page 3Linked to original sources

arXiv2019

Multiplicity of the saturated special fiber ring of height two perfect ideals

Let $R$ be a polynomial ring and $I \subset R$ be a perfect ideal of height two minimally generated by forms of the same degree. We provide a formula for the multiplicity of the saturated special fiber ring of $I$. Interestingly, this formula is equal to an elementary symmetric polynomial in terms of the degrees of the syzygies of $I$. Applying ideas introduced in arXiv:1805.05180, we obtain the value of the j-multiplicity of $I$ and an effective method for determining the degree and birationality of rational maps defined by homogeneous generators of $I$.

math.AC↗

arXiv2018

Bounding the degrees of a minimal $μ$-basis for a rational surface parametrization

In this paper, we study how the degrees of the elements in a minimal $μ$-basis of a parametrized surface behave. For an arbitrary rational surface parametrization $P(s,t)=(a_1(s,t),a_2(s,t),a_3(s,t),a_4(s,t)) \in \mathbb{F}[s,t]^4$ over an infinite field $\mathbb{F}$, we show the existence of a $μ$-basis with polynomials bounded in degree by $O(d^{33})$, where $d=\max(\text{deg}(a_1),\text{deg}(a_2), \text{deg}(a_3), \text{deg}(a_4))$. Under additional assumptions we can obtain tighter bounds.

math.AC↗

arXiv2018

Regularity of bicyclic Graphs and their powers

Let $I(G)$ be the edge ideal of a bicyclic graph. In this paper, we characterize the Castelnuovo-Mumford regularity of $I(G)$ in terms of the induced matching number of $G$. For the base case of this family of graphs, i.e. dumbbell graph, we explicitly compute the induced matching number. Moreover, we prove that $ \operatorname{reg}(I(G)^q)=2q+\operatorname{reg}(I(G))-2$, for all $ q\geq 1 $, when $ G $ is a dumbbell graph with a connecting path having no more than two vertices.

math.AC↗

arXiv2018

A $D$-module approach on the equations of the Rees algebra

Let $I \subset R = \mathbb{F}[x_1,x_2]$ be a height two ideal minimally generated by three homogeneous polynomials of the same degree $d$, where $\mathbb{F}$ is a field of characteristic zero. We use the theory of $D$-modules to deduce information about the defining equations of the Rees algebra of $I$. Let $\mathcal{K}$ be the kernel of the canonical map $α: \text{Sym}(I) \rightarrow \text{Rees}(I)$ from the symmetric algebra of $I$ onto the Rees algebra of $I$. We prove that $\mathcal{K}$ can be described as the solution set of a system of differential equations, that the whole bigraded structure of $\mathcal{K}$ is characterized by the integral roots of certain $b$-functions, and that certain de Rham cohomology groups can give partial information about $\mathcal{K}$.

math.AC↗

arXiv2018

Regularity and Gröbner bases of the Rees algebra of edge ideals of bipartite graphs

Let $G$ be a bipartite graph and $I=I(G)$ be its edge ideal. The aim of this note is to investigate different aspects of the Rees algebra $\mathcal{R}(I)$ of $I$. We compute its regularity and the universal Gröbner basis of its defining equations; interestingly, both of them are described in terms of the combinatorics of $G$. We apply these ideas to study the regularity of the powers of $I$. For any $s \ge \text{match}(G)+\lvert E(G) \rvert +1$ we prove that $\text{reg}(I^{s+1})=\text{reg}(I^s)+2$.

math.AC↗