Mixed radix numeration bases: Horner's rule, Yang-Baxter equation and Furstenberg's conjecture

TL;DR

This paper explores the use of mixed radix bases in number theory, introducing algorithms for basis conversion related to Horner's rule, and connects these to the Yang-Baxter equation, offering new insights into Furstenberg's conjecture.

Contribution

It introduces a novel connection between mixed radix basis conversions, the Yang-Baxter equation, and Furstenberg's conjecture, with algorithms based on two-dimensional arrays.

Findings

Algorithms for basis conversion using Euclidean division

Relation between basis transformations and the Yang-Baxter equation

Reinterpretation of Furstenberg's conjecture through Yang-Baxter transformations

Abstract

Mixed radix bases in numeration is a very old notion but it is rarely studied on its own or in relation with concrete problems related to number theory. Starting from the natural question of the conversion of a basis to another for integers as well as polynomials, we use mixed radix bases to introduce two-dimensional arrays with suitable filling rules. These arrays provide algorithms of conversion which uses only a finite number of euclidean division to convert from one basis to another; it is interesting to note that these algorithms are generalizations of the well-known Horner's rule of quick evaluation of polynomials. The two-dimensional arrays with local transformations are reminiscent from statistical mechanics models: we show that changes between three numeration basis are related to the set-theoretical Yang-Baxter equation and this is, up to our knowledge, the first time that such a structure is described in number theory. As an illustration, we reinterpret well-known results around Furstenberg's conjecture in terms of Yang-Baxter transformations between mixed radix bases, hence opening the way to alternative approaches.