Polynomials that Sign Represent Parity and Descartes’ Rule of Signs

Abstract.

A real polynomial P(X ₁, ... , X _n ) sign represents f : A ⁿ → {0, 1} if for every (a ₁, ... , a _n ) ∈ A ⁿ, the sign of P(a ₁, ... ,a _n ) equals $(-1)^{f(a_{1}, \ldots ,a_{n})}$. Such sign representations are well-studied in computer science and have applications to computational complexity and computational learning theory. The work in this area aims to determine the minimum degree and sparsity possible for a polynomial that sign represents a function f. While the degree of such polynomials is relatively well-understood, far less is known about their sparsity. Known bounds apply only to the cases where A = {0, 1} or A = {−1, +1}.

In this work, we present a systematic study of tradeoffs between degree and sparsity of sign representations through the lens of the parity function. We attempt to prove bounds that hold for any choice of set A. We show that sign representing parity over {0,... , m − 1}ⁿ with the degree in each variable at most m − 1 requires sparsity at least m ⁿ. We show that a tradeoff exists between sparsity and degree, by exhibiting a sign representation that has higher degree but lower sparsity. We show a lower bound of n(m − 2) + 1 on the sparsity of polynomials of any degree representing parity over {0,... ,m − 1}ⁿ. We prove exact bounds on the sparsity of such polynomials for any two element subset A. The main tool used is Descartes’ Rule of Signs, a classical result in algebra, relating the sparsity of a polynomial to its number of real roots. As an application, we use bounds on sparsity to derive circuit lower bounds for depth-two AND-OR-NOT circuits with a Threshold Gate at the top.