The Recovery of $\lambda$ from a Hilbert Polynomial

TL;DR

This paper presents an efficient algorithm to determine if a polynomial is a Hilbert polynomial and to recover the associated integer partition, aiding geometric and combinatorial analysis of Hilbert schemes.

Contribution

The paper introduces a novel algorithm that efficiently identifies Hilbert polynomials and reconstructs the corresponding integer partition, advancing computational methods in algebraic geometry.

Findings

Algorithm accurately recovers partitions from Hilbert polynomials

Method improves efficiency over previous approaches

Facilitates analysis of geometric properties in Hilbert schemes

Abstract

In the study of Hilbert schemes, the integer partition $\lambda$ helps researchers identify some geometric and combinatorial properties of the scheme in question. To aid researchers in extracting such information from a Hilbert polynomial, we describe an efficient algorithm which can identify if $p(x)\in\mathbb{Q}[x]$ is a Hilbert polynomial and if so, recover the integer partition $\lambda$ associated with it.