Maximums of Total Betti Numbers in Hilbert Families
Mathematics, Statistics, and Computer Science
Betti numbers, Hilbert function, commutative algebra, free resolutions, lexicographic order, polynomial ring
Fix a family of ideals in a polynomial ring and consider the problem of finding a single ideal in the family that has Betti numbers that are greater than or equal to the Betti numbers of every ideal in the family. Or decide if this special ideal even exists. Bigatti, Hulett, and Pardue showed that if we take the ideals with a fixed Hilbert function, there is such an ideal: the lexsegment ideal. Caviglia and Murai proved that if we take the saturated ideals with a fixed Hilbert polynomial, there is also such an ideal. We present a generalization of these two situations, an algorithm for determining the existence of these special ideals and finding them when they do exist, and some cases where we guarantee existence.
White, J. (2021). Maximums of Total Betti Numbers in Hilbert Families. https://doi.org/10.13023/etd.2021.187