On $q$-deformed cubic equations: the quantum heptagon and nonagon

TL;DR

This paper extends the concept of $q$-deformed irrational numbers to cubic equations, specifically those related to regular 7- and 9-gons, revealing new invariance properties under modular group actions.

Contribution

It introduces a canonical $q$-deformation for certain cubic equations with cyclic Galois groups, expanding the theory beyond quadratic irrationals.

Findings

Cubic equations with cyclic Galois group $C_3$ have a canonical $q$-deformation.

Includes examples related to regular 7- and 9-gons.

Shows invariance under specific modular group actions.

Abstract

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.