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

Hervé Fournier; Sylvain Perifel; Rémi de Verclos

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arXiv · 1304.5910

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

Hervé Fournier·Sylvain Perifel·Rémi de Verclos

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Abstract

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.

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BibTeX RIS

Hervé Fournier, Sylvain Perifel, Rémi de Verclos. 2013-04-22. On fixed-polynomial size circuit lower bounds for uniform polynomials in the sense of Valiant. https://arxiv.org/abs/1304.5910

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