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# On Polynomial Approximations Over Z/2^kZ*

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LIPIcs.STACS.2017.12.pdf
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## Cite As

Abhishek Bhrushundi, Prahladh Harsha, and Srikanth Srinivasan. On Polynomial Approximations Over Z/2^kZ*. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 12:1-12:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.STACS.2017.12

## Abstract

We study approximation of Boolean functions by low-degree polynomials over the ring Z/2^kZ. More precisely, given a Boolean function F:{0,1}^n -> {0,1}, define its k-lift to be F_k:{0,1}^n -> {0,2^(k-1)} by F_k(x) = 2^(k-F(x)) (mod 2^k). We consider the fractional agreement (which we refer to as \gamma_{d,k}(F)) of F_k with degree d polynomials from Z/2^kZ[x_1,..,x_n]. Our results are the following: * Increasing k can help: We observe that as k increases, gamma_{d,k}(F) cannot decrease. We give two kinds of examples where gamma_{d,k}(F) actually increases. The first is an infinite family of functions F such that gamma_{2d,2}(F) - gamma_{3d-1,1}(F) >= Omega(1). The second is an infinite family of functions F such that gamma_{d,1}(F) <= 1/2+o(1) - as small as possible - but gamma_{d,3}(F) >= 1/2 + Omega(1). * Increasing k doesn't always help: Adapting a proof of Green [Comput. Complexity, 9(1):16--38, 2000], we show that irrespective of the value of k, the Majority function Maj_n satisfies gamma_{d,k}(Maj_n) <= 1/2+ O(d)/sqrt{n}. In other words, polynomials over Z/2^kZ for large k do not approximate the majority function any better than polynomials over Z/2Z. We observe that the model we study subsumes the model of non-classical polynomials, in the sense that proving bounds in our model implies bounds on the agreement of non-classical polynomials with Boolean functions. In particular, our results answer questions raised by Bhowmick and Lovett [In Proc. 30th Computational Complexity Conf., pages 72-87, 2015] that ask whether non-classical polynomials approximate Boolean functions better than classical polynomials of the same degree.
##### Keywords
• Polynomials over rings
• Approximation by polynomials
• Boolean functions
• Non-classical polynomials

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