Testing polynomials which are easy to compute (Extended Abstract)

We exploit the fact that the set of all polynomials Pε@@@@[x₁,.,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 a¹,.,a^sε [u]ⁿ = {1,2,.,u}ⁿ such that for all polynomials PεW(d,n,v):P(a¹) = p(a²) =...= p(a^s) = O implies PΞO. This means that a¹,...,a^s is a correct test sequence for a zero test on all polynomials in W(d,n,v). Moreover, “almost every” sequence a¹,.,a^sε[u]ⁿ is such a correct test sequence for W(d,n,v). The existence of correct test sequences a¹,.,a^sε [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,.,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.