Nutan Limaye: Superpolynomial lower bounds against low-depth algebraic circuits

Joint BARC/MIAO seminar Friday Nov 12, 2021
Superpolynomial lower bounds against low-depth algebraic circuits
(Nutan Limaye, IT University of Copenhagen)

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.

In the first part of this two-part talk, I will present the statements of the main results and some background. In the second part of the talk, I will give some proof details.

This is a joint work with Srikanth Srinivasan and Sébastien Tavenas.
For more information about the MIAO seminars, please see http://www.jakobnordstrom.se/videoseminars/ .

Видео Nutan Limaye: Superpolynomial lower bounds against low-depth algebraic circuits канала MIAO Research