On the complexity of partial derivatives

TL;DR

This paper investigates the computational complexity of determining the dimension of the partial derivatives space of a polynomial, establishing its #P-hardness, and explores the trace method for lower bounds in algebraic complexity.

Contribution

It proves the #P-hardness of computing the partial derivatives dimension and analyzes the trace method as a polynomial-time lower bound technique.

Findings

Computing the dimension is #P-hard.

The trace method yields polynomial-time lower bounds.

Open problem: approximation algorithms with guarantees.

Abstract

The method of partial derivatives is one of the most successful lower bound methods for arithmetic circuits. It uses as a complexity measure the dimension of the span of the partial derivatives of a polynomial. In this paper, we consider this complexity measure as a computational problem: for an input polynomial given as the sum of its nonzero monomials, what is the complexity of computing the dimension of its space of partial derivatives? We show that this problem is #P-hard and we ask whether it belongs to #P. We analyze the "trace method", recently used in combinatorics and in algebraic complexity to lower bound the rank of certain matrices. We show that this method provides a polynomial-time computable lower bound on the dimension of the span of partial derivatives, and from this method we derive closed-form lower bounds. We leave as an open problem the existence of an approximation algorithm with reasonable performance guarantees.A slightly shorter version of this paper was presented at STACS'17. In this new version we have corrected a typo in Section 4.1, and added a reference to Shitov's work on tensor rank.