On the bit-size of non-radical triangular sets

TL;DR

This paper establishes upper bounds on the bit-size of coefficients in non-radical triangular Groebner bases of zero-dimensional ideals, extending previous radical set results and introducing smaller coefficient polynomials.

Contribution

It extends bounds to non-radical triangular sets and introduces a family of smaller polynomials, addressing limitations of previous height-based methods.

Findings

Bounds depend on input data due to multiplicities

Introduces smaller coefficient polynomials

Extends previous radical set results to non-radical sets

Abstract

We present upper bounds on the bit-size of coefficients of non-radical lexicographical Groebner bases in purely triangular form (triangular sets) of dimension zero. This extends a previous work [Dahan-Schost, Issac'2004], constrained to radical triangular sets; it follows the same technical steps, based on interpolation. However, key notion of height of varieties is not available for points with multiplicities; therefore the bounds obtained are less universal and depend on some input data. We also introduce a related family of non- monic polynomials that have smaller coefficients, and smaller bounds. It is not obvious to compute them from the initial triangular set though.