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Computing Generic Fibers of Polynomial Ideals with FGLM and Hensel Lifting

Jérémy Berthomieu, Rafael Mohr

TL;DR

A version of the FGLM algorithm that can be used to compute generic fibers of positive-dimensional polynomial ideals and it is shown that this algorithm has a complexity quasi-linear in the number of terms of certain <Formula format="inline"><TexMath><?TeX $\mathfrak {m}$?></TexMath><AltText>Math 1</AltText>-adic expansions.

Abstract

We describe a version of the FGLM algorithm that can be used to compute generic fibers of positive-dimensional polynomial ideals. It combines the FGLM algorithm with a Hensel lifting strategy. In analogy with Hensel lifting, we show that this algorithm has a complexity quasi-linear in the number of terms of certain $\mathfrak{m}$-adic expansions we compute. Some provided experimental data also demonstrates the practical efficacy of our algorithm.

Computing Generic Fibers of Polynomial Ideals with FGLM and Hensel Lifting

TL;DR

A version of the FGLM algorithm that can be used to compute generic fibers of positive-dimensional polynomial ideals and it is shown that this algorithm has a complexity quasi-linear in the number of terms of certain <Formula format="inline"><TexMath><?TeX ?></TexMath><AltText>Math 1</AltText>-adic expansions.

Abstract

We describe a version of the FGLM algorithm that can be used to compute generic fibers of positive-dimensional polynomial ideals. It combines the FGLM algorithm with a Hensel lifting strategy. In analogy with Hensel lifting, we show that this algorithm has a complexity quasi-linear in the number of terms of certain -adic expansions we compute. Some provided experimental data also demonstrates the practical efficacy of our algorithm.
Paper Structure (10 sections, 8 theorems, 26 equations, 2 algorithms)

This paper contains 10 sections, 8 theorems, 26 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Scientific Context
  3. Problem Statement & Contributions
  4. Related Work
  5. Outline
  6. Preliminaries
  7. Gröbner Bases
  8. Points of Good Specialization
  9. The Main Algorithm
  10. Complexity Estimates

Key Result

theorem 1

Let $f_1,\ldots,f_c$ be generic polynomials of respective degrees $d_1,\ldots,d_c$ in $\mathbb{K}[\mathbf{z},\mathbf{x}]$. Assume that the $\prec_{\mathop{\mathrm{drl}}\nolimits}$-Gröbner basis of $I=\langle f_1,\ldots,f_c\rangle$ is known and that the $\prec_{\mathop{\mathrm{out}}\nolimits}$-Gröbne

Theorems & Definitions (32)

  • theorem 1
  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • definition 5
  • definition 6
  • definition 7
  • definition 8
  • definition 9
  • ...and 22 more