Title: Testing polynomials which are easy to compute (Extended Abstract)
Abstract: We exploit the fact that the set of all polynomials Pε@@@@[x1,..,xn] 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 a1,..,asε [u]n = {1,2,..,u}n such that for all polynomials PεW(d,n,v):P(a1) = p(a2) =...= p(as) = O implies PΞO. This means that a1,...,as is a correct test sequence for a zero test on all polynomials in W(d,n,v). Moreover, "almost every" sequence a1,..,asε[u]n is such a correct test sequence for W(d,n,v). The existence of correct test sequences a1,..,asε [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@@@@[x1,..,xn] with deg(P) = d and small integer coefficients such that [email protected]@@@ W(d,n,v). For v>n log d lower bounds of this type are beyond our present methods in algebraic complexity theory.