Stronger Lower Bounds and Randomness-Hardness Trade-Offs Using Associated Algebraic Complexity Classes

We associate to each Boolean language complexity class C the algebraic class a·C consisting of families of polynomials {fn} for which the evaluation problem over Z is in C. We prove the following lower bound and randomness-to-hardness results: 1. If polynomial identity testing (PIT) is in NSUBEXP then a·NEXP does not have poly size constant-free arithmetic circuits. 2. a·NEXP does not have poly size constant-free arithmetic circuits. 3. For every fixed k, a·MA does not have arithmetic circuits of size n. Items 1 and 2 strengthen two results due to Kabanets and Impagliazzo [6]. The third item improves a lower bound due to Santhanam [11]. We consider the special case low-PIT of identity testing for (constant-free) arithmetic circuits with low formal degree, and give improved hardness-to-randomness trade-offs that apply to this case. Combining our results for both directions of the hardness-randomness connection, we demonstrate a case where derandomization of PIT and proving lower bounds are equivalent. Namely, we show that low-PIT ∈ i.o-NTIME[2no(1) ]/n if and only if there exists a family of multilinear polynomials in a·NE/lin that requires constant-free arithmetic circuits of super-polynomial size and formal degree. ∗Supported by EPSRC Grant H05068X/1 †Supported in part by EPSRC Grant H05068X/1