On Read-Once Functions over $\boldsymbol{\mathbb{Z}}_{\mathbf{3}}$

Abstract

We consider read-once functions over $\mathbb{Z}_{3}$, i.e., functions expressed by a formula with addition and multiplication modulo $3$ and constants 0, 1, 2 that contains every variable at most once. For a function $F(x_{1},\dots,x_{n})\colon\{0,1,2\}^{n}\to\{0,1,2\}$ let $w_{i}$, $i=0,1,2$, be the number of tuples $(\sigma_{1},\dots,\sigma_{n})\in\{0,1,2\}^{n}$ such that $F(\sigma_{1},\dots,\sigma_{n})=i$ and let $p_{i}=w_{i}/3^{n}$. We prove that for every read-once function the values $p_{0}$, $p_{1}$, $p_{2}$ satisfy the inequality $\max p_{i}-\min p_{i}\leq(\max p_{i}+\min p_{i})^{3}$, and that this bound is in a certain sense sharp for read-once functions over $\mathbb{Z}_{3}$.