Quasi-Stable ideals and Borel-fixed ideals with a given Hilbert Polynomial

TL;DR

This paper develops algorithms to classify all saturated quasi-stable and Borel-fixed ideals in polynomial rings with a specified Hilbert polynomial, using combinatorial structures like Pommaret bases.

Contribution

It introduces two algorithms for enumerating saturated quasi-stable and Borel-fixed ideals with a given Hilbert polynomial, leveraging Pommaret bases for their combinatorial structure.

Findings

Algorithms successfully list all saturated quasi-stable ideals for given Hilbert polynomial.

Algorithms enumerate all saturated Borel-fixed ideals considering field characteristic.

Utilizes Pommaret bases to analyze the combinatorial structure of the ideals.

Abstract

The present paper investigates properties of quasi-stable ideals and of Borel-fixed ideals in a polynomial ring $k[x_0,\dots,x_n]$, in order to design two algorithms: the first one takes as input $n$ and an admissible Hilbert polynomial $P(z)$, and outputs the complete list of saturated quasi-stable ideals in the chosen polynomial ring with the given Hilbert polynomial. The second algorithm has an extra input, the characteristic of the field $k$, and outputs the complete list of saturated Borel-fixed ideals in $k[x_0,\dots,x_n]$ with Hilbert polynomial $P(z)$. The key tool for the proof of both algorithms is the combinatorial structure of a quasi-stable ideal, in particular we use a special set of generators for the considered ideals, the Pommaret basis.