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 {f1,...,fm} ⊂ double-struck F[x1,...x1] 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(fi)}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 f1,...,fm 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 ∑Π∑Π δ(k,s,n) circuit C to be of the form ∈i=1k Πj=1s fi,j, where f i,j are sparse n-variate polynomials of degree at most δ. For k = 2, we give a poly(δsn)δ2 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 ∑Π∑Πδ(k,s,n) identities, we give a poly(δsnR)Rkδ2 time blackbox identity test for ∑Π∑Πδ(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. © 2011 Springer-Verlag.