Arithmetic circuit lower bounds from sumset expansion

2026-07-17

AI Summary

This work addresses a central barrier to resolving Valiant’s conjecture (VP ≠ VNP) by explicitly constructing elusive functions with prescribed parameters. By restricting coordinate mappings to monomials and leveraging hitting sets formed from roots of unity together with Chebotarev’s theorem, the construction of elusive functions is reduced— for the first time—to a problem of exponential sumset expansion in additive combinatorics. This approach not only resolves an open question posed by Garg et al. regarding the existence of elusive curves of exponential degree but also yields an explicit construction of such curves. Furthermore, for arithmetic circuits of depth below \(o(\log n / \log \log n)\), the method improves the best-known superlinear lower bound on input size to a quadratic one.

Abstract

Raz proposed a program to prove arithmetic circuit lower bounds through the explicit construction of elusive functions. These are polynomial maps from a low dimensional space to a high dimensional ambient space whose image is contained in no subvariety of low complexity. Here, complexity is prescribed in terms of the dimension and degree of parametric maps into the ambient space defining the subvariety. Elusive functions are abundant: finding explicit ones with parameters typical of generic polynomial maps implies Valiant's hypothesis that VP$\neq$VNP. But no such construction is known. Raz devised elusive functions with weaker parameters to derive explicit degree d polynomials in n variables requiring superlinear circuit size at depth $d=o(\log n)$. We present a new method to analyse and construct elusive functions, with coordinate maps restricted to monomials. To prove elusiveness, we identify a hitting set of points, each a tuple of roots of unity coupled based on the exponents of the monomial maps. Using Chebotarev's theorem on roots of unity, we show that for every low complexity subvariety, the function evaluated at some point in the hitting set eludes it. For this strategy to work, it suffices that the iterated sumset of a certain set of numbers (derived from the exponents) expands exponentially. We thus reduce open explicit construction problems in elusive functions to purely additive combinatorial ones, whose resolutions imply as yet unknown lower bounds. Informed by iterated sumset expansion, we devise new elusive functions. We construct explicit elusive curves of exponential degree, resolving an open problem posed by Garg, Makam, Oliveira, and Wigderson as a testament to the difficulty of elusiveness proofs. We improve Raz's superlinear bound quadratically (with circuit size to input size ratio as the metric) below $o(\log n/\log\log n)$ depths.

Problem

• elusive functions
• arithmetic circuit lower bounds
• explicit construction
• sumset expansion
• VP vs VNP

Innovation

• elusive functions
• sumset expansion
• arithmetic circuit lower bounds
• hitting set
• Chebotarev's theorem