Algebraic independence and blackbox identity testing

Abstract

Algebraic independence is a fundamental notion in commutative algebra that generalizes independence of linear polynomials. Polynomials {f1,…,fm}⊂K[x1,…,xn] (over a field K) are called algebraically independent if there is no non-zero polynomial F such that F(f1,…,fm)=0. The transcendence degree, trdeg{f1,…,fm}, is the maximal number r of algebraically independent polynomials in the set. 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(fi)}i=r, assuming sparse fi and small r. We apply these fundamental maps to solve two cases of blackbox identity testing (assuming a large or zero characteristic):1.Given a polynomial-degree circuit C and sparse polynomials f1,…,fm of transcendence degree r, we can test blackbox D:=C(f1,…,fm) for zeroness in poly(size(D))r time.2.Define a ΣΠΣΠδ(k,s,n) circuit to be of the form ∑i=1k∏j=1sfi,j, where fi,j are sparse n-variate polynomials of degree at most δ. For this class of depth-4 circuits we define a notion of rank. Assuming there is a rank bound R for minimal simple ΣΠΣΠδ(k,s,n) identities, we give a poly(δsnR)Rkδ2 time blackbox identity test for ΣΠΣΠδ(k,s,n) circuits. This partially generalizes the state of the art of depth-3 to depth-4 circuits. The notion of transcendence degree works best with large or zero characteristic, but we also give versions of our results for arbitrary fields.