Abstract

https://youtu.be/8B4qZ6KLdqA

A large number of algebraic problems can be reduced to the problem of computing families of polynomials. We conventionally model these computations using arithmetic circuits. The central question in the subject, an algebraic analogue of the P vs NP problem, asks whether there is a super-polynomial lower bound on the sizes of circuits computing the permanent. We introduce symmetric arithmetic circuits, i.e. arithmetic circuits with a natural symmetry restriction. In the context of circuits computing polynomials defined on a matrix of variables, such as the determinant or the permanent, the restriction amounts to requiring that the shape of the circuit is invariant under row and column permutations of the matrix. We establish unconditional exponential, lower bounds on the size of any symmetric circuit for computing the permanent over any field of characteristic other than 2. In contrast, we show that there are polynomial-size symmetric circuits for computing the determinant over fields of characteristic zero.