Special-case Algorithms for Blackbox Radical Membership, Nullstellensatz and Transcendence Degree

TL;DR

This paper presents specialized algorithms for radical membership, Nullstellensatz, and transcendence degree problems that are efficient when the transcendence degree is smaller than the number of variables, improving upon prior bounds.

Contribution

It introduces algorithms with runtime around d^r for radical membership and Nullstellensatz when the transcendence degree r is less than n, along with improved degree bounds.

Findings

Radical membership testing in about d^r time for small r

Improved Nullstellensatz degree bounds when r is much less than n

Reduction of problems to cases with r+1 polynomials of high transcendence degree

Abstract

Radical membership testing, and the special case of Hilbert's Nullstellensatz (HN), is a fundamental computational algebra problem. It is NP-hard; and has a famous PSPACE algorithm due to effective Nullstellensatz bounds. We identify a useful case of these problems where practical algorithms, and improved bounds, could be given, when the transcendence degree $r$ of the input polynomials is smaller than the number of variables $n$. If $d$ is the degree bound on the input polynomials, then we solve radical membership (even if input polynomials are blackboxes) in around $d^r$ time. The prior best was $> d^n$ time (always, $d^n\ge d^r$). Also, we significantly improve effective Nullstellensatz degree-bound, when $r\ll n$. Structurally, our proof shows that these problems reduce to the case of $r+1$ polynomials of transcendence degree $\ge r$. This input instance (corresponding to none or a unique annihilator) is at the core of HN's hardness. Our proof methods invoke basic algebraic-geometry.