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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.19798 |
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| _version_ | 1866909554532417536 |
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| author | Simon, Damien |
| author_facet | Simon, Damien |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_19798 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mixed radix numeration bases: Horner's rule, Yang-Baxter equation and Furstenberg's conjecture Simon, Damien Mathematical Physics Statistical Mechanics Number Theory Probability Exactly Solvable and Integrable Systems 11A63, 11A67, 16T25 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. |
| title | Mixed radix numeration bases: Horner's rule, Yang-Baxter equation and Furstenberg's conjecture |
| topic | Mathematical Physics Statistical Mechanics Number Theory Probability Exactly Solvable and Integrable Systems 11A63, 11A67, 16T25 |
| url | https://arxiv.org/abs/2405.19798 |