# Cores of Monomial Ideals

The other classes of ideals which we consider are zero-dimensional monomial ideals in the polynomial ring R=k[x_1,...,x_d]. We show that, if such an ideal I is an almost complete intersection, then the core of I satisfies a (d+1)-fold symmetry property coming from the generators of I, and hence that the shape of core(I) is closely related to the shape of I. Using this symmetry property, we completely describe the shape of core(I) in the case where I has a monomial minimal reduction. Then, in the two-dimensional case, we give an algorithm for computing the core of I which allows us to prove that the minimal number of generators of core(I) is 2r+2, where r is the reduction number of I. We prove that this result holds even for ideals which do not have a minimal reduction which is monomial. We also describe the core of an ideal I in k[x,y] having more generators, in the case where I has a monomial minimal reduction. Finally, we consider a strongly stable ideal I in k[x,y] having a monomial minimal reduction J. We prove that r_J(I) leq mu(I)-2, and we give an algorithm for obtaining the core of I via its first coefficient ideal.

## History

## Date Modified

2017-06-02## Defense Date

2010-03-31## Research Director(s)

Claudia Polini## Committee Members

Yu Xie Nero Budur Juan Migliore## Degree

- Doctor of Philosophy

## Degree Level

- Doctoral Dissertation

## Language

- English

## Alternate Identifier

etd-04162010-172451## Publisher

University of Notre Dame## Program Name

- Mathematics