Algebraic Independence and Blackbox Identity Testing

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 ∑k i=1 ∏s j=1 fi,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) Rkδ 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.