TCS+ Talk

Wednesday October 13, 2021 10:00 AM

Superpolynomial Lower Bounds Against Low-Depth Algebraic Circuits

Speaker: Nutan Limaye, Computer Science and Engineering, IT University of Copenhagen
Location: Online Event

Abstract: Every multivariate polynomial P(X) can be written as a sum of monomials, i.e. a sum of products of variables and field constants. In general, the size of such an expression is the number of monomials that have a non-zero coefficient in P.

What happens if we add another layer of complexity, and consider sums of products of sums (of variables and field constants) expressions? Now, it becomes unclear how to prove that a given polynomial P(X) does not have small expressions. In this result, we solve exactly this problem.

More precisely, we prove that certain explicit polynomials have no polynomial-sized "Sigma-Pi-Sigma" (sums of products of sums) representations. We can also show similar results for Sigma-Pi-Sigma-Pi, Sigma-Pi-Sigma-Pi-Sigma and so on for all "constant-depth" expressions.  

The talk is based on a joint work of Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas.

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