On $q$-deformed cubic equations: the quantum heptagon and nonagon
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916368932143104 |
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| author | Ovsienko, Valentin Ustinov, Alexey |
| author_facet | Ovsienko, Valentin Ustinov, Alexey |
| contents | The recent notion of $q$-deformed irrational numbers is characterized by the invariance with respect to the action of the modular group $\PSL(2,\Z)$, or equivalently under the Burau representation of the braid group~$B_3$. The theory of $q$-deformed quadratic irrationals and quadratic equations with integer coefficients is known and entirely based on this invariance. In this paper, we consider the case of cubic irrationals. We show that irreducible cubic equations with three distinct real roots and cyclic Galois group~$C_3$ (or $\Z/3\Z$) acting by a third order element of $\PSL(2,\Z)$, have a canonical $q$-deformation, that we describe. This class of cubic equations contains well-known examples including the equations that describe regular $7$- and $9$-gons. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_13670 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On $q$-deformed cubic equations: the quantum heptagon and nonagon Ovsienko, Valentin Ustinov, Alexey Combinatorics The recent notion of $q$-deformed irrational numbers is characterized by the invariance with respect to the action of the modular group $\PSL(2,\Z)$, or equivalently under the Burau representation of the braid group~$B_3$. The theory of $q$-deformed quadratic irrationals and quadratic equations with integer coefficients is known and entirely based on this invariance. In this paper, we consider the case of cubic irrationals. We show that irreducible cubic equations with three distinct real roots and cyclic Galois group~$C_3$ (or $\Z/3\Z$) acting by a third order element of $\PSL(2,\Z)$, have a canonical $q$-deformation, that we describe. This class of cubic equations contains well-known examples including the equations that describe regular $7$- and $9$-gons. |
| title | On $q$-deformed cubic equations: the quantum heptagon and nonagon |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2408.13670 |