Algebraic independence and blackbox identity testing

• Authors: M. Beecken;J. Mittmann;N. Saxena
• Venue: Information and Computation
• Year: 2013

Algebraic independence is a fundamental notion in commutative algebra that generalizes independence of linear polynomials. Polynomials {f"1,...,f"m}@?K[x"1,...,x"n] (over a field K) 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 @f that reduce the number of variables from n to r but maintain trdeg{@f(f"i)}"i=r, assuming sparse f"i 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 f"1,...,f"m of transcendence degree r, we can test blackbox D:=C(f"1,...,f"m) for zeroness in poly(size(D))^r time. 2.Define a @S@P@S@P"@d(k,s,n) circuit to be of the form @?"i"="1^k@?"j"="1^sf"i","j, where f"i","j are sparse n-variate polynomials of degree at most @d. For this class of depth-4 circuits we define a notion of rank. Assuming there is a rank bound R for minimal simple @S@P@S@P"@d(k,s,n) identities, we give a poly(@dsnR)^R^k^@d^^^2 time blackbox identity test for @S@P@S@P"@d(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.