  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^k) over the complex field (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^k). Then we investigate the link between
fixed-polynomial size circuit bounds in the Boolean and arithmetic settings. In
characteristic zero, it is proved that $\NP \not\subset \size(n^k)$, or $\MA
\subset \size(n^k)$, or NP=MA imply lower bounds on the circuit size of uniform
polynomials in n variables from the class VNP over the complex field, assuming
GRH. In positive characteristic p, uniform polynomials in VNP have circuits of
fixed-polynomial size if and only if both VP=VNP over F_p and Mod_pP has
circuits of fixed-polynomial size.
