2024/05/30 by Damien Simon, Simon, Damien · 1 citation
Computer Science · Mathematics · #11A63 #11A67 #16T25 #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Approximation and Integration #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Number Theory (math.NT) #Numerical Methods and Algorithms #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech)
paper · pdf · doi:10.48550/arxiv.2405.19798
openalex publication_date 2024/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.