Interpolation in Valiant's theory

TL;DR

This paper explores the relationship between boolean algorithms evaluating polynomials at rational points and the existence of polynomial-size arithmetic circuits, linking algebraic and boolean complexity classes through interpolation methods.

Contribution

It introduces a novel approach using Lagrange interpolation to connect boolean evaluation of polynomials with algebraic circuit complexity, and establishes new implications for Valiant's hypothesis.

Findings

Boolean evaluation of polynomials relates to algebraic circuit size

Constant-free VP vs VNP hypothesis implications

Efficient computation of exponential sums impacts P=NP in BSS model

Abstract

We investigate the following question: if a polynomial can be evaluated at rational points by a polynomial-time boolean algorithm, does it have a polynomial-size arithmetic circuit? We argue that this question is certainly difficult. Answering it negatively would indeed imply that the constant-free versions of the algebraic complexity classes VP and VNP defined by Valiant are different. Answering this question positively would imply a transfer theorem from boolean to algebraic complexity. Our proof method relies on Lagrange interpolation and on recent results connecting the (boolean) counting hierarchy to algebraic complexity classes. As a byproduct we obtain two additional results: (i) The constant-free, degree-unbounded version of Valiant's hypothesis that VP and VNP differ implies the degree-bounded version. This result was previously known to hold for fields of positive characteristic only. (ii) If exponential sums of easy to compute polynomials can be computed efficiently, then the same is true of exponential products. We point out an application of this result to the P=NP problem in the Blum-Shub-Smale model of computation over the field of complex numbers.