A hitting set construction, with application to arithmetic circuit lower bounds

TL;DR

This paper presents a deterministic identity testing algorithm for specific univariate polynomials and uses it to establish exponential lower bounds for certain polynomial representations, advancing the understanding of arithmetic circuit complexity.

Contribution

It introduces a new derandomized identity testing algorithm for a class of univariate polynomials and demonstrates its application to proving lower bounds for polynomial representations.

Findings

Deterministic black-box identity testing algorithm for specific polynomials

Exponential lower bounds for polynomial representations like rac{X^i-1}{X}

Feasibility of hardness-from-derandomization approach for restricted circuits

Abstract

A polynomial identity testing algorithm must determine whether a given input polynomial is identically equal to 0. We give a deterministic black-box identity testing algorithm for univariate polynomials of the form $\sum_{j=0}^t c_j X^{\alpha_j} (a + b X)^{\beta_j}$. From our algorithm we derive an exponential lower bound for representations of polynomials such as $\prod_{i=1}^{2^n} (X^i-1)$ under this form. It has been conjectured that these polynomials are hard to compute by general arithmetic circuits. Our result shows that the "hardness from derandomization" approach to lower bounds is feasible for a restricted class of arithmetic circuits. The proof is based on techniques from algebraic number theory, and more precisely on properties of the height function of algebraic numbers.