The $q$-analog of the Markoff injectivity conjecture over the language of a balanced sequence

TL;DR

This paper proves a $q$-analog of the Markoff injectivity conjecture for balanced sequences, showing that a certain polynomial map is injective over their language, extending previous conjectures to a broader context.

Contribution

It establishes the injectivity of a $q$-analog map on balanced sequences, linking combinatorics on words with Markoff properties and polynomial inequalities.

Findings

Existence of an order making the map injective

Polynomials with nonnegative coefficients for differences

Injectivity extends to the language of balanced sequences

Abstract

The Markoff injectivity conjecture states that $w\mapsto\mu(w)_{12}$ is injective on the set of Christoffel words where $\mu:\{\mathtt{0},\mathtt{1}\}^*\to\mathrm{SL}_2(\mathbb{Z})$ is a certain homomorphism and $M_{12}$ is the entry above the diagonal of a $2\times2$ matrix $M$. Recently, Leclere and Morier-Genoud (2021) proposed a $q$-analog $\mu_q$ of $\mu$ such that $\mu_{q}(w)_{12}|_{q=1}=\mu(w)_{12}$ is the Markoff number associated to the Christoffel word $w$ when evaluated at $q=1$. We show that there exists an order $<_{radix}$ on $\{\mathtt{0},\mathtt{1}\}^*$ such that for every balanced sequence $s \in \{\mathtt{0},\mathtt{1}\}^\mathbb{Z}$ and for all factors $u, v$ in the language of $s$ with $u <_{radix} v$, the difference $\mu_q(v)_{12} - \mu_q(u)_{12}$ is a nonzero polynomial of indeterminate $q$ with nonnegative integer coefficients. Therefore, the map $u\mapsto\mu_q(u)_{12}$ is injective over the language of a balanced sequence. The proof uses an equivalence between balanced sequences satisfying some Markoff property and indistinguishable asymptotic pairs.