Contribution in combinatorics in commutative algebra.(ph-d thesis)

TL;DR

This thesis advances combinatorics in commutative algebra by introducing new classes of monomial ideals, computing generic initial ideals for specific algebraic structures, and proving cases of Moreno's conjecture.

Contribution

It introduces $eta$-fixed ideals generalizing $p$-Borel ideals and extends results to this class, also proving Moreno's conjecture for certain cases.

Findings

Defined $eta$-fixed ideals generalizing $p$-Borel ideals

Computed generic initial ideals of complete intersections with the strong Lefschetz property

Proved Moreno's conjecture for $n=3$ in characteristic zero

Abstract

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 to this new class. In the second chapter, we compute the generic initial ideal, with repect to the reverse lexicographic order, of an ideal which define a complete intersection of embedding dimension three with strong Lefschetz property and we show that it is an almost reverse lexicographic ideal. This enable us to give a proof for Moreno's conjecture in the case $n=3$ and characteristic zero. Also, we prove that the $d$-component of the generic initial ideal, with respect to the reverse lexicographic order, of an ideal generated by a regular sequence of homogeneous polynomials of degree $d$ is revlex, in a particular, but important, case.