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

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.

75 pages

References in corpus (8)

• Castelnuovo-Mumford regularity of products of ideals
• Generic initial ideals of points and curves
• The strong Lefschetz property and simple extensions
• Generic initial ideal for complete intersections of embedding dimension three with strong Lefschetz property
• A stable property of Borel type ideals
• Monomial ideals with linear upper bound regularity
• Regularity for certain classes of monomial ideals
• A note on the generic initial ideal for complete intersections