A promenade through Correct Test Sequences I: Degree of constructible sets, B\'ezout's Inequality and density

TL;DR

This paper explores the foundational aspects and generalizations of correct test sequences, demonstrating their widespread occurrence, developing degree theory for constructible sets, and providing algorithms and bounds related to polynomial identity testing.

Contribution

It generalizes the existence and density of short correct test sequences for constructible sets, introduces a degree theory for these sets, and connects these concepts to probabilistic algorithms and classical polynomial bounds.

Findings

Existence of dense, short correct test sequences in constructible sets

Development of a degree theory for constructible sets with Bezout's inequalities

A BPP algorithm for polynomial list verification using correct test sequences

Abstract

In Heintz-Schnorr (1982), the authors introduced the notion of correct test sequence and since then it has been widely used to design probabilistic algorithms for Polynomial Equality Test. The aim of this manuscript is to study the foundations and generalizations of this notion. We show that correct test sequences are almost omnipresent and appear in many different forms in the mathematical literature: As identity sequences for Function Identity Test, as norming sets in the field of Banach algebras or as samples in the context of Reproducing Kernel Hilbert Spaces. As main outcome, we generalize the main statement of Heintz-Schnorr (1982) proving that short correct test sequences for constructible sets of lists of polynomials do exist and are densely distributed in any constructible set of accurate dimension and degree. The main tool used to prove this result is the theory of degree of constructible sets, which we introduce and develop in this manuscript, generalizing the results of Heintz (1983) and proving two Bezout's Inequalities for two different notions of degree. We present a ${\bf BPP}_K$ algorithm to exhibit the power of correct test sequences, this algorithm decides whether a list of polynomials is a secant sequence by just evaluating the input list at some well-suited points. We show the differences between correct test sequences and Demillo-Lipton-Schwartz-Zippel probabilistic tests and we reformulate, prove and generalize two well-known results of the Polynomial Method: We prove Dvir's exponential lower bounds for Kakeya sets from lower bounds for the length of correct test sequences and generalize Alon's Combinatorial Nullstellensatz.