On the relative power of reduction notions in arithmetic circuit complexity

TL;DR

This paper demonstrates that p-projections and c-reductions differ in power within arithmetic circuit complexity by showing certain polynomials are VNP-complete under c-reductions but not p-projections, with results depending on the field.

Contribution

It establishes the fundamental difference in power between p-projections and c-reductions in arithmetic circuit complexity and explores the field dependence of VNP-completeness.

Findings

Existence of polynomials VNP-complete under c-reductions but not p-projections

Unconditional separation of reduction notions

Field dependence of VNP-completeness

Abstract

We show that the two main reduction notions in arithmetic circuit complexity, p-projections and c-reductions, differ in power. We do so by showing unconditionally that there are polynomials that are VNP-complete under c-reductions but not under p-projections. We also show that the question of which polynomials are VNP-complete under which type of reductions depends on the underlying field.