A PSPACE Construction of a Hitting Set for the Closure of Small Algebraic Circuits

TL;DR

This paper presents a PSPACE algorithm for constructing a hitting set that works for the closure of VP, enabling deterministic solutions for algebraic problems previously solvable only with randomized or higher complexity methods.

Contribution

It introduces a PSPACE construction of a hitting set for the closure of VP, improving upon prior exponential space methods and enabling deterministic solutions for related algebraic problems.

Findings

Constructed a PSPACE algorithm for hitting sets in algebraic circuit closures.

Demonstrated deterministic solutions for algebraic problems like Noether Normalization in PSPACE.

Introduced the concept of robust hitting sets and proved their existence using algebraic geometry and anti-concentration.

Abstract

In this paper we study the complexity of constructing a hitting set for the closure of VP, the class of polynomials that can be infinitesimally approximated by polynomials that are computed by polynomial sized algebraic circuits, over the real or complex numbers. Specifically, we show that there is a PSPACE algorithm that given n,s,r in unary outputs a set of n-tuples over the rationals of size poly(n,s,r), with poly(n,s,r) bit complexity, that hits all n-variate polynomials of degree-r that are the limit of size-s algebraic circuits. Previously it was known that a random set of this size is a hitting set, but a construction that is certified to work was only known in EXPSPACE (or EXPH assuming the generalized Riemann hypothesis). As a corollary we get that a host of other algebraic problems such as Noether Normalization Lemma, can also be solved in PSPACE deterministically, where earlier only randomized algorithms and EXPSPACE algorithms (or EXPH assuming the generalized Riemann hypothesis) were known. The proof relies on the new notion of a robust hitting set which is a set of inputs such that any nonzero polynomial that can be computed by a polynomial size algebraic circuit, evaluates to a not too small value on at least one element of the set. Proving the existence of such a robust hitting set is the main technical difficulty in the proof. Our proof uses anti-concentration results for polynomials, basic tools from algebraic geometry and the existential theory of the reals.