Arithmetic Circuits: A Chasm at Depth 3

SIAM J. Comput., Volume 45, Issue 3, 2016, Pages 1064-1079.

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Abstract:

We show that, over ${mathbb Q}$, if an $n$-variate polynomial of degree $d = n^{O(1)}$ is computable by an arithmetic circuit of size $s$ (respectively, by an arithmetic branching program of size $s$), then it can also be computed by a depth-3 circuit (i.e., a $Sigma Pi Sigma$ circuit) of size $exp(O(sqrt{d log n log dlog s}))$ (respectiv...More

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