## Abstract

A real multivariate polynomial *p*(*x*
_{1}, …, *x*
_{
n
}) is said to *sign-represent* a Boolean function *f*: {0,1}^{n}→{−1,1} if the sign of *p*(*x*) equals *f*(*x*) for all inputs *x*∈{0,1}^{n}. We give new upper and lower bounds on the degree of polynomials which sign-represent Boolean functions. Our upper bounds for Boolean formulas yield the first known subexponential time learning algorithms for formulas of *superconstant* depth. Our lower bounds for constant-depth circuits and intersections of halfspaces are the first new degree lower bounds since 1968, improving results of Minsky and Papert. The lower bounds are proved *constructively*; we give explicit dual solutions to the necessary linear programs.

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## Additional information

A preliminary version of these results appeared as [24].

This work was done while at the Department of Mathematics, MIT, Cambridge, MA, and while supported by NSF grant 99-12342.

Supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship and by NSF grant CCR-98-77049. This work was done while at the Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA.

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O’Donnell, R., Servedio, R.A. New degree bounds for polynomial threshold functions.
*Combinatorica* **30, **327–358 (2010). https://doi.org/10.1007/s00493-010-2173-3

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### Mathematics Subject Classification (2000)

- 68Q17
- 68Q32