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The Recovery of $λ$ from a Hilbert Polynomial

Joseph Donato, Monica Lewis

TL;DR

Problem: Given a polynomial $p(x)\in \mathbb{Q}[x]$, determine whether it is a Hilbert polynomial and recover the associated partition $\lambda$ describing its Hilbert decomposition. Approach: contrast a naive interpolation-based recovery with a robust method built on discrete derivatives, binomial sequences, and a reduction procedure. Contributions: (i) explicit Hilbert-polynomial criterion $p(x)=\sum_i \binom{x+\lambda_i-i}{\lambda_i-1}$; (ii) a discrete-derivative recovery algorithm that peels off layers via $\Delta$ and binomial terms; (iii) complexity analysis showing improved worst-case performance over the naive approach. Impact: provides a practical tool for identifying Hilbert polynomials and extracting $\lambda$-data relevant to Hilbert schemes and related geometric properties.

Abstract

In the study of Hilbert schemes, the integer partition $λ$ 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 $λ$ associated with it.

The Recovery of $λ$ from a Hilbert Polynomial

TL;DR

Problem: Given a polynomial , determine whether it is a Hilbert polynomial and recover the associated partition describing its Hilbert decomposition. Approach: contrast a naive interpolation-based recovery with a robust method built on discrete derivatives, binomial sequences, and a reduction procedure. Contributions: (i) explicit Hilbert-polynomial criterion ; (ii) a discrete-derivative recovery algorithm that peels off layers via and binomial terms; (iii) complexity analysis showing improved worst-case performance over the naive approach. Impact: provides a practical tool for identifying Hilbert polynomials and extracting -data relevant to Hilbert schemes and related geometric properties.

Abstract

In the study of Hilbert schemes, the integer partition 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 is a Hilbert polynomial and if so, recover the integer partition associated with it.
Paper Structure (5 sections, 6 theorems, 11 equations, 3 algorithms)

This paper contains 5 sections, 6 theorems, 11 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Naive Algorithm
  3. Discrete Derivatives
  4. Discrete Derivative Algorithm
  5. Conclusions

Key Result

Theorem 1

Mac Let $R=\mathbb C[x_{0},...,x_{n}]$ and $p(x)\in\mathbb Q[x]$, there exists ideals in $R$ with Hilbert polynomial $p(x)$ iff $p(x)$ can be written in the form $\sum_{i=1}^{r}\binom{x+\lambda_{i}-i}{\lambda_{i}-1}$ for some integer partition $\lambda=(\lambda_1,...,\lambda_r)$ where $n\geq\lambda_

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • Theorem 4
  • proof
  • Theorem 5
  • Theorem 6
  • proof