Algebraic Independence and Blackbox Identity Testing

Abstract

Algebraic independence is an advanced notion in commutative algebra that generalizes independence of linear polynomials to higher degree. The transcendence degree (trdeg) of a set $\{f_1, \dotsc, f_m\} \subset \mathbb{F}[x_1, \dotsc, x_n]$ of polynomials is the maximal size r of an algebraically independent subset. In this paper we design blackbox and efficient linear maps ϕ that reduce the number of variables from n to r but maintain trdeg{ϕ(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 circuit C and sparse subcircuits f ₁,…,f _m of trdeg r such that D: = C(f ₁,…,f _m ) has polynomial degree, we can test blackbox D for zeroness in poly(size(D))^r time.

• 2 Define a ΣΠΣΠ _δ (k,s,n) circuit C to be of the form ∑  _i = 1 ^k ∏  _j = 1 ^s f _i,j , where f _i,j are sparse n-variate polynomials of degree at most δ. For k = 2, we give a $poly(\delta sn)^{\delta^2}$ 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 ΣΠΣΠ _δ (k,s,n) identities, we give a $poly(\delta snR)^{Rk\delta^2}$ time blackbox identity test for ΣΠΣΠ _δ (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.

The first two authors are grateful to the Bonn International Graduate School in Mathematics for research funding.