关于$q$-变形三次方程：量子七边形与九边形
On $q$-deformed cubic equations: the quantum heptagon and nonagon
AI总结：
本文研究三次无理数，证明了具有三个不同实根且Galois群为$C_3$的三次不可约方程存在典范$q$-变形，并指出正七边形和九边形方程属于此类。
摘要：
近期提出的$q$-变形无理数概念，其特征在于对模群$\PSL(2,\Z)$作用的不变性，或等价地，在辫群~$B_3$的Burau表示下的不变性。$q$-变形二次无理数及具有整数系数的二次方程的理论已为人知，且完全基于此不变性。本文探讨三次无理数的情形。我们证明：具有三个不同实根且Galois群~$C_3$（或$\Z/3\Z$）通过$\PSL(2,\Z)$的三阶元素作用的三次不可约方程，存在典范的$q$-变形，并对其进行了描述。此类三次方程包含著名例子，即描述正$7$边形和$9$边形的方程。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
