Tavenas (Proceedings of mathematical foundations of computer science (MFCS), 2013) has recently proved that any nO(1)\documentclass[12pt]{minimal}
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\begin{document}$$n^{O(1)}$$\end{document}-variate and degree n polynomial in VP\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {VP}$$\end{document} can be computed by a depth-4
ΣΠΣΠ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma \Pi \Sigma \Pi $$\end{document}
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\begin{document}$$2^{O(\sqrt{n}\log n)}$$\end{document}. So, to prove VP≠VNP\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {VP}\ne \mathsf {VNP}$$\end{document} it is sufficient to show that an explicit polynomial in VNP\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {VNP}$$\end{document} of degree n requires 2ω(nlogn)\documentclass[12pt]{minimal}
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\begin{document}$$2^{\omega (\sqrt{n}\log n)}$$\end{document} size depth-4 circuits. Soon after Tavenas’ result, for two different explicit polynomials, depth-4 circuit-size lower bounds of 2Ω(nlogn)\documentclass[12pt]{minimal}
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\begin{document}$$2^{\Omega (\sqrt{n}\log n)}$$\end{document} have been proved (see Kayal et al. in Proceedings of symposium on theory of computing, ACM, 2014b. http://doi.acm.org/10.1145/2591796.2591847; Fournier et al. in Proceedings of symposium on theory of computing, ACM, 2014). In particular, using a combinatorial design Kayal et al. (2014b) construct an explicit polynomial in VNP\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {VNP}$$\end{document} that requires depth-4 circuits of size 2Ω(nlogn)\documentclass[12pt]{minimal}
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\begin{document}$$2^{\Omega (\sqrt{n}\log n)}$$\end{document} and Fournier et al. (Proceedings of symposium on theory of computing, ACM, 2014) show that the iterated matrix multiplication polynomial (which is in VP\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {VP}$$\end{document}) also requires 2Ω(nlogn)\documentclass[12pt]{minimal}
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\begin{document}$$2^{\Omega (\sqrt{n}\log n)}$$\end{document} size depth-4 circuits.
