A <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> </mml:math> -analog of the Markoff injectivity conjecture holds
Sebastien Labbé, Mélodie Lapointe, Wolfgang Steiner
Abstract
The elements of Markoff triples are given by coefficients in certain matrix products defined by Christoffel words, and the Markoff injectivity conjecture, a longstanding open problem (also known as the uniqueness conjecture), is then equivalent to injectivity on Christoffel words. A <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> </mml:math> -analog of these matrix products has been proposed recently, and we prove that injectivity on Christoffel words holds for this <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> </mml:math> -analog. The proof is based on the evaluation at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mo form="prefix">exp</mml:mo> <mml:mo>(</mml:mo> <mml:mi>i</mml:mi> <mml:mi>π</mml:mi> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Other roots of unity provide some information on the original problem, which corresponds to the case <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . We also extend the problem to arbitrary words and provide a large family of pairs of words where injectivity does not hold.