Algebraic Independence and Blackbox Identity Testing

TL;DR

This paper introduces algebraic independence-based methods and linear maps to improve blackbox identity testing for various classes of polynomials and circuits, achieving more efficient algorithms under certain conditions.

Contribution

It develops new algebraic tools and reductions that enable faster blackbox identity testing for complex polynomial circuits, extending previous results to depth-4 circuits.

Findings

Poly-time blackbox zero testing for certain polynomial circuits.

New identity tests for depth-4 circuits with rank assumptions.

Effective reduction of variables preserving algebraic independence.

Abstract

Algebraic independence is an advanced notion in commutative algebra that generalizes independence of linear polynomials to higher degree. Polynomials {f_1, ..., f_m} \subset \F[x_1, ..., x_n] are called algebraically independent if there is no non-zero polynomial F such that F(f_1, ..., f_m) = 0. The transcendence degree, trdeg{f_1, ..., f_m}, is the maximal number r of algebraically independent polynomials in the set. In this paper we design blackbox and efficient linear maps \phi that reduce the number of variables from n to r but maintain trdeg{\phi(f_i)}_i = r, assuming f_i's sparse and small r. We apply these fundamental maps to solve several cases of blackbox identity testing: (1) Given a polynomial-degree circuit C and sparse polynomials f_1, ..., f_m with trdeg r, we can test blackbox D := C(f_1, ..., f_m) for zeroness in poly(size(D))^r time. (2) Define a spsp_\delta(k,s,n) circuit C to be of the form \sum_{i=1}^k \prod_{j=1}^s f_{i,j}, where f_{i,j} are sparse n-variate polynomials of degree at most \delta. For k = 2 we give a poly(sn\delta)^{\delta^2} time blackbox identity test. (3) For a general depth-4 circuit we define a notion of rank. Assuming there is a rank bound R for minimal simple spsp_\delta(k,s,n) identities, we give a poly(snR\delta)^{Rk\delta^2} time blackbox identity test for spsp_\delta(k,s,n) circuits. This partially generalizes the state of the art of depth-3 to depth-4 circuits. The notion of trdeg works best with large or zero characteristic, but we also give versions of our results for arbitrary fields.