# Probabilistic Polynomials and Hamming Nearest Neighbors

@article{Alman2015ProbabilisticPA, title={Probabilistic Polynomials and Hamming Nearest Neighbors}, author={Josh Alman and Ryan Williams}, journal={2015 IEEE 56th Annual Symposium on Foundations of Computer Science}, year={2015}, pages={136-150} }

We show how to compute any symmetric Boolean function on n variables over any field (as well as '/ the integers) with a probabilistic polynomial of degree O( √nlog(1/ε)) and error at most ε. The degree dependence on n and ε is optimal, matching a lower bound of Razborov (1987) and Smolensky (1987) for the MAJORITY function. The proof is constructive: a low-degree polynomial can be efficiently sampled from the distribution. This polynomial construction is combined with other algebraic ideas to… CONTINUE READING

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