STOC Conference 1980 Conference Paper
- Joos Heintz
- Claus-Peter Schnorr
We exploit the fact that the set of all polynomials Pε@@@@[x 1 ,.,x n ] of degree ≤d which can be evaluated with ≤v nonscalar steps can be embedded into a Zariski-closed affine set W(d,n,v),dim W(d,n,v)≤(v+1 +n) 2 and deg W(d,n,v)≤(2vd) (v+1+n) 2 . As a consequence we prove that for u:= 2v(d+1) 2 and s:= 6(v+1+n) 2 there exist a 1 ,., a s ε [u] n = {1,2,.,u} n such that for all polynomials PεW(d,n,v):P( a 1 ) = p( a 2 ) =...= p( a s ) = O implies PΞO. This means that a 1 ,..., a s is a correct test sequence for a zero test on all polynomials in W(d,n,v). Moreover, “almost every” sequence a 1 ,., a s ε[u] n is such a correct test sequence for W(d,n,v). The existence of correct test sequences a 1 ,., a s ε [u] n is established by a counting argument without constructing a correct test sequence. We even show that it is beyond the known methods to establish (i.e. to construct and to prove correctness) of such a short correct test sequence for W(d,n,v). We prove that given such a short, correct test sequence for W(d,n,v) we can efficiently construct a multivariate polynomial Pε@@@@[x 1 ,.,x n ] with deg(P) = d and small integer coefficients such that P@@@@ W(d,n,v). For v>n log d lower bounds of this type are beyond our present methods in algebraic complexity theory.