# Depth-4 Lower Bounds, Determinantal Complexity: A Unified Approach

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## Abstract

Tavenas (Proceedings of mathematical foundations of computer science (MFCS), 2013) has recently proved that any \(n^{O(1)}\)-variate and degree *n* polynomial in \(\mathsf {VP}\) can be computed by a depth-4
\(\Sigma \Pi \Sigma \Pi \)
circuit of size \(2^{O(\sqrt{n}\log n)}\). So, to prove \(\mathsf {VP}\ne \mathsf {VNP}\) it is sufficient to show that an explicit polynomial in \(\mathsf {VNP}\) of degree *n* requires \(2^{\omega (\sqrt{n}\log n)}\) size depth-4 circuits. Soon after Tavenas’ result, for two different explicit polynomials, depth-4 circuit-size lower bounds of \(2^{\Omega (\sqrt{n}\log n)}\) 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 \(\mathsf {VNP}\) that requires depth-4 circuits of size \(2^{\Omega (\sqrt{n}\log n)}\) and Fournier et al. (Proceedings of symposium on theory of computing, ACM, 2014) show that the iterated matrix multiplication polynomial (which is in \(\mathsf {VP}\)) also requires \(2^{\Omega (\sqrt{n}\log n)}\) size depth-4 circuits.

In this paper, we identify a simple combinatorial property such that any polynomial *f* that satisfies this property would achieve a similar depth-4 circuit-size lower bound. In particular, it does not matter whether *f* is in \(\mathsf {VP}\) or in \(\mathsf {VNP}\). As a result, we get a simple unified lower-bound analysis for the above-mentioned polynomials.

Another goal of this paper is to compare our current knowledge of the depth-4 circuit-size lower bounds and the determinantal complexity lower bounds. Currently, the best known determinantal complexity lower bound is \(\Omega (n^2)\) for permanent of a \(n\times n\) matrix (which is a \(n^2\)-variate and degree *n* polynomial) due to Cai et al. (Proceedings of symposium on theory of computing, ACM, 2008). We prove that the determinantal complexity of the iterated matrix multiplication polynomial is \(\Omega (dn)\) where *d* is the number of matrices and *n* is the dimension of the matrices. In particular, our result settles the determinantal complexity of the iterated matrix multiplication polynomial to \(\Theta (dn)\). To the best of our knowledge, a \(\Theta (n)\) bound for the determinantal complexity for the iterated matrix multiplication polynomial was known only for any constant \(d>1\), due to Jansen (Theory Comput Syst 49(2):343–354, 2011).

## Keywords

Lower bounds Determinantal complexity Constant depth Arithmetic circuits## Subject classification

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## Notes

### Acknowledgements

This work was done when Suryajith Chillara was a graduate student at Chennai Mathematical Institute. Suryajith Chillara was supervised by Partha Mukhopadhyay and was supported by TCS research fellowship. We thank the anonymous reviewers for their invaluable feedback that helped improve the paper and take the current form.

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