Sharp upper bounds of the Betti numbers for a given Hilbert polynomial

TL;DR

This paper establishes sharp upper bounds on the Betti numbers of saturated graded ideals in polynomial rings for a fixed Hilbert polynomial, identifying ideals that achieve these bounds.

Contribution

It introduces the existence of saturated graded ideals with maximal Betti numbers for any given Hilbert polynomial, providing a precise extremal bound.

Findings

Existence of saturated ideals with maximal Betti numbers.

Explicit bounds for Betti numbers based on Hilbert polynomial.

Identification of extremal ideals achieving these bounds.

Abstract

We show that there exists a saturated graded ideal in a standard graded polynomial ring which has the largest total Betti numbers among all saturated graded ideals for a fixed Hilbert polynomial.