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Conditions of positivity on a shadow Markoff Tree

Nathan Bonin

TL;DR

This work analyzes positivity conditions for solutions of the shadow Markoff equation $A^2+B^2+C^2=(3-\sigmaoldsymbol{ abla}oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{}}}}}}}}}}}})}ABC$ with dual numbers $A=a+oldsymbol{ extepsilon}oldsymbol{ extalpha}$, $B=b+oldsymbol{ extepsilon}oldsymbol{ extbeta}$, $C=c+oldsymbol{ extepsilon}oldsymbol{ extgamma}$ and $oldsymbol{ extepsilon}^2=0$. The author formulates a conjecture that positivity along mutations is governed by the rational point $(oldsymbol{ extalpha}:oldsymbol{ extbeta}:oldsymbol{ extgamma})$ lying in a convex polygon $P$ in ${ m RP}^2$, whose vertices are $(0:0:1),(1:0:2),(1:1:1),(0:2:1)$. A key theoretical advance is the linear dependence of the nilpotent part $oldsymbol{ extalpha}'$ on $(oldsymbol{ extalpha},oldsymbol{ extbeta},oldsymbol{ extgamma})$, enabling a convex-geometry analysis that underpins a main theorem tying positivity to membership in $P$. The paper then constructs four shadow Markoff trees corresponding to the polygon’s vertices, linking shadow sequences to classical integer sequences (e.g., Fibonacci, Pell) and providing Sage-based numerical evidence that supports the conjectured polygon as the positivity domain. Overall, the work offers a convex-geometric framework for positivity in shadow Markoff dynamics and connects this framework to classic Markoff theory and combinatorial sequences.

Abstract

An analogue of the Markoff equation has recently been introduced by the author and Valentin Ovsienko. A conjecture about the necessary and sufficient conditions for positivity of solutions to this equation is formulated and discussed.

Conditions of positivity on a shadow Markoff Tree

TL;DR

This work analyzes positivity conditions for solutions of the shadow Markoff equation with dual numbers , , and . The author formulates a conjecture that positivity along mutations is governed by the rational point lying in a convex polygon in , whose vertices are . A key theoretical advance is the linear dependence of the nilpotent part on , enabling a convex-geometry analysis that underpins a main theorem tying positivity to membership in . The paper then constructs four shadow Markoff trees corresponding to the polygon’s vertices, linking shadow sequences to classical integer sequences (e.g., Fibonacci, Pell) and providing Sage-based numerical evidence that supports the conjectured polygon as the positivity domain. Overall, the work offers a convex-geometric framework for positivity in shadow Markoff dynamics and connects this framework to classic Markoff theory and combinatorial sequences.

Abstract

An analogue of the Markoff equation has recently been introduced by the author and Valentin Ovsienko. A conjecture about the necessary and sufficient conditions for positivity of solutions to this equation is formulated and discussed.
Paper Structure (13 sections, 3 theorems, 20 equations)

This paper contains 13 sections, 3 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. The classic Markoff tree
  3. Proof of the main theorem
  4. Four shadow Markoff trees
  5. The tree of the vertex $(0:0:1)$
  6. The tree of the vertex $(1:0:2)$
  7. The case of the vertex $(1:1:1)$: double Markoff tree
  8. The tree of the vertex $(0:2:1)$
  9. Code and numeric evidence for the conjecture
  10. Top side
  11. Bottom side
  12. Up side
  13. Acknowledgments

Key Result

Theorem 1

Every triple leading to a positive integer tree of solutions to SMarEq can be obtained from some initial triple InitTrip with the point $(\alpha:\beta:\gamma)\in\mathbb{Q}$P$^2$ that belongs to some convex polygon by a series of mutations and permutations along the Markoff tree.

Theorems & Definitions (6)

  • Conjecture 1
  • Theorem 1
  • Theorem 2
  • proof
  • Lemma 3
  • proof