On Fixed-Polynomial Size Circuit Lower Bounds for Uniform Polynomials in the Sense of Valiant

We consider the problem of fixed-polynomial lower bounds on the size of arithmetic circuits computing uniform families of polynomials. Assuming the generalised Riemann hypothesis (GRH), we show that for all k, there exist polynomials with coefficients in MA having no arithmetic circuits of size O(n) over C (allowing any complex constant). We also build a family of polynomials that can be evaluated in AM having no arithmetic circuits of size O(n). Then we investigate the link between fixed-polynomial size circuit bounds in the Boolean and arithmetic settings. In characteristic zero, it is proved that NP 6⊂ size(n), or MA ⊂ size(n), or NP = MA imply lower bounds on the circuit size of uniform polynomials in n variables from the class VNP over C, assuming GRH. In positive characteristic p, uniform polynomials in VNP have circuits of fixed-polynomial size if and only if both VP = VNP over Fp and ModpP has circuits of fixed-polynomial size.