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孙晓明

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期刊论文

On the Degree of Boolean Functions as Polynomials over Zm

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arXiv,2020,(): | 2020年05月01日 | arXiv:1910.12458

URL:https://arxiv.org/abs/1910.12458?context=cs

摘要/描述

Polynomial representations of Boolean functions over various rings such as Z and Zm have been studied since Minsky and Papert (1969). From then on, they have been employed in a large variety of fields including communication complexity, circuit complexity, learning theory, coding theory and so on. For any integer m≥2, each Boolean function has a unique multilinear polynomial representation over ring Zm. The degree of such polynomial is called modulo-m degree, denoted as degm(⋅). In this paper, we investigate the lower bound of modulo-m degree of Boolean functions. When m=pk (k≥1) for some prime p, we give a tight lower bound that degm(f)≥k(p−1) for any non-degenerated function f:{0,1}n→{0,1}, provided that n is sufficient large. When m contains two different prime factors p and q, we give a nearly optimal lower bound for any symmetric function f:{0,1}n→{0,1} that degm(f)≥n2+1p−1+1q−1.

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