Affine projections of symmetric polynomials

摘要

In this paper, we introduce a new model for computing polynomials—a depth-2 circuit with a symmetric gate at the top and plus gates at the bottom, i.e. the circuit computes a symmetric function in linear functions—Smd(ℓ1,ℓ2,…,ℓm)(Smd is the dth elementary symmetric polynomial in m variables, and the ℓi's are linear functions). We refer to this model as the symmetric model. This new model is related to standard models of arithmetic circuits, especially to depth-3 circuits. In particular, we show that in order to improve the results of Shpilka and Wigderson (in: CCC, Vol. 14, 1999, pp. 87–96), i.e. to prove super-quadratic lower bounds for depth-3 circuits, one must first prove a super-linear lower bound for the symmetric model.We prove two non-trivial linear lower bounds for our model. The first lower bound is for computing the determinant, and the second is for computing the sum of two monomials. The main technical contribution relates the maximal dimension of linear subspaces on which Smd vanishes to lower bounds in the symmetric model. In particular, we show that an answer of the following problem (which is very natural, and of independent interest) will imply lower bounds on symmetric circuits for many polynomials:What is the maximal dimension of a linear subspace of Cm, on which Smd vanishes?We give two partial solutions to the problem above, each enables us to prove a different lower bound. Using our techniques we also prove quadratic lower bounds for depth-3 circuits computing the elementary symmetric polynomials of degree αn (where 0<α<1 is a constant), thus extending the result of Shpilka and Wigderson (in: CCC, Vol. 14, 1999, pp. 87–96). These are the best lower bounds known for depth-3 circuits over fields of characteristic zero.