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Assuming the Generalised Riemann Hypothesis (GRH), we show that for all k,there exist polynomials with coefficients in $\MA$ having no arithmeticcircuits of size O(n^k) over the complex field (allowing any complex constant).We also build a family of polynomials that can be evaluated in AM having noarithmetic circuits of size O(n^k). Then we investigate the link betweenfixed-polynomial size circuit bounds in the Boolean and arithmetic settings. Incharacteristic zero, it is proved that $\NP \not\subset \size(n^k)$, or $\MA\subset \size(n^k)$, or NP=MA imply lower bounds on the circuit size of uniformpolynomials in n variables from the class VNP over the complex field, assumingGRH. In positive characteristic p, uniform polynomials in VNP have circuits offixed-polynomial size if and only if both VP=VNP over F_p and Mod_pP hascircuits of fixed-polynomial size.

On fixed-polynomial size circuit lower bounds for uniform polynomials in the sense of Valiant