On the arity gap of polynomial functions

TL;DR

This paper characterizes polynomial functions with high arity gaps over various fields, providing explicit decomposition schemes for finite fields and identifying limitations for infinite fields of odd characteristic.

Contribution

It offers a detailed description of polynomial functions with arity gap at least 3 and those with gap 2 over different fields, refining previous results.

Findings

Explicit decomposition schemes for polynomial functions over finite fields.

Polynomial functions with arity gap ≥ 3 characterized.

Limitations identified for infinite fields of odd characteristic.

Abstract

The authors' previous results on the arity gap of functions of several variables are refined by considering polynomial functions over arbitrary fields. We explicitly describe the polynomial functions with arity gap at least 3, as well as the polynomial functions with arity gap equal to 2 for fields of characteristic 0 or 2. These descriptions are given in the form of decomposition schemes of polynomial functions. Similar descriptions are given for arbitrary finite fields. However, we show that these descriptions do not extend to infinite fields of odd characteristic.