Degree-uniform lower bound on the weights of polynomials with given sign function

Vladimir V. Podolskii

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algorithmic aspects of algebra and logic, Tome 274 (2011), pp. 252-268

Résumé

A Boolean function $f\colon\{0,1\}^n\to\{0,1\}$ is called the sign function of an integer polynomial $p$ of degree $d$ in $n$ variables if it is true that $f(x)=1$ if and only if $p(x)>0$. In this case the polynomial $p$ is called a threshold gate of degree $d$ for the function $f$. The weight of the threshold gate is the sum of the absolute values of the coefficients of $p$. For any $n$ and $d\le D\le\frac{\varepsilon n^{1/5}}{\log n}$ we construct a function $f$ such that there is a threshold gate of degree $d$ for $f$, but any threshold gate for $f$ of degree at most $D$ has weight $2^{(\delta n)^d/D^{4d}}$, where $\varepsilon>0$ and $\delta>0$ are some constants. In particular, if $D$ is constant, then any threshold gate of degree $D$ for our function has weight $2^{\Omega(n^d)}$. Previously, functions with these properties have been known only for $d=1$ (and arbitrary $D$) and for $D=d$. For constant $d$ our functions are computable by polynomial size DNFs. The best previous lower bound on the weights of threshold gates for such functions was $2^{\Omega(n)}$. Our results can also be translated to the case of functions $f\colon\{-1,1\}^n\to\{-1,1\}$.