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

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## Abstract

A Boolean function *f*: {0, 1}^{ n } → {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* ≤ *D* ≤ \(\frac{{\varepsilon n^{1/5} }}
{{\log n}}
\)
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 *ɛ* > 0 and *δ* > 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^{Ω(n)}. Our results can also be translated to the case of functions *f*: {−1, 1}^{ n } → {−1, 1}.

## Keywords

STEKLOV Institute Boolean Function Maximal Element Ordinal Number Disjunctive Normal Form## References

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