Special-case Algorithms for Blackbox Radical Membership, Nullstellensatz and 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.