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The tropical critical point and mirror symmetry

Jamie Judd, Konstanze Rietsch

TL;DR

The paper proves that a complete, positive Laurent polynomial $W$ over the field of generalised Puiseux series $oldsymbol{K}$ has a unique positive critical point $p_{ ext{crit}}\, ext{in}\,T(oldsymbol{K}_{>0})$, and assigns to it a canonical tropical critical point $d_{ ext{crit}}$ via valuation. It introduces an augmented Newton polytope framework to characterize the maximal value of Trop$(W)$ and develops a recursive construction for $d_{ ext{crit}}$, along with a leading-term analysis yielding a unique leading coefficient $d_{ ext{coeff}}$. The work extends to Puiseaux generalisations, rationality, and nondegeneracy, and connects to toric geometry by producing canonical non-displaceable Lagrangian tori from the mirror potential, with compatibility under mutations so that the canonical data persist across cluster-variety mutations. The results unify aspects of tropical geometry, Puiseux-analytic methods, and mirror symmetry for toric and non-toric settings, providing tools to extract canonical geometric objects from a family of Laurent polynomials. The approach yields concrete geometric consequences, including central Lagrangian fibers in toric manifolds and orbifolds, and suggests a mutation-invariant perspective on canonical mirrors.

Abstract

Call a Laurent polynomial $W$ `complete' if its Newton polytope is full-dimensional with zero in its interior. We show that if $W$ is any complete Laurent polynomial with coefficients in the positive part of the field $K$ of generalised Puiseux series, then $W$ has a unique positive critical point $p_{crit}$. Here a generalised Puiseux series is called `positive' if the coefficient of its leading term is in $\mathbb R_{>0}$. Using the valuation on $K$ we obtain a canonically associated `tropical critical point' $d_{crit}$ in $\mathbb R^{r}$ for which we give a finite recursive construction. We show that this result is compatible with a general form of mutation, so that it can be applied in a cluster varieties setting. We also give applications to toric geometry including, via the theory of [FOOO], to the construction of canonical non-displaceable Lagrangian tori for toric symplectic manifolds.

The tropical critical point and mirror symmetry

TL;DR

The paper proves that a complete, positive Laurent polynomial over the field of generalised Puiseux series has a unique positive critical point , and assigns to it a canonical tropical critical point via valuation. It introduces an augmented Newton polytope framework to characterize the maximal value of Trop and develops a recursive construction for , along with a leading-term analysis yielding a unique leading coefficient . The work extends to Puiseaux generalisations, rationality, and nondegeneracy, and connects to toric geometry by producing canonical non-displaceable Lagrangian tori from the mirror potential, with compatibility under mutations so that the canonical data persist across cluster-variety mutations. The results unify aspects of tropical geometry, Puiseux-analytic methods, and mirror symmetry for toric and non-toric settings, providing tools to extract canonical geometric objects from a family of Laurent polynomials. The approach yields concrete geometric consequences, including central Lagrangian fibers in toric manifolds and orbifolds, and suggests a mutation-invariant perspective on canonical mirrors.

Abstract

Call a Laurent polynomial `complete' if its Newton polytope is full-dimensional with zero in its interior. We show that if is any complete Laurent polynomial with coefficients in the positive part of the field of generalised Puiseux series, then has a unique positive critical point . Here a generalised Puiseux series is called `positive' if the coefficient of its leading term is in . Using the valuation on we obtain a canonically associated `tropical critical point' in for which we give a finite recursive construction. We show that this result is compatible with a general form of mutation, so that it can be applied in a cluster varieties setting. We also give applications to toric geometry including, via the theory of [FOOO], to the construction of canonical non-displaceable Lagrangian tori for toric symplectic manifolds.

Paper Structure

This paper contains 39 sections, 37 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Positivity for tori and tropicalisation
  4. Positivity and the field of generalised Puiseux series
  5. Algebraic tori over $\mathbf K$ and tropicalisation
  6. Leading terms and exponential map for $T(\mathbf K_{>0})$
  7. The positive critical point theorem
  8. Proof of Theorem \ref{['t:main']} and Theorem \ref{['t:critmax']}
  9. The augmented Newton polytope and the maximum value of $\operatorname{Trop}(W)$.
  10. The tropical critical conditions
  11. Claim:
  12. Proof of Claim:
  13. Claim:
  14. Proof of the Claim:
  15. Induction hypothesis:
  16. ...and 24 more sections

Key Result

Theorem 1.1

Let $W=\sum \gamma_i x^{v_i}$ be a Laurent polynomial satisfying the following two properties. Then $W$ has a unique critical point in $(\mathbf K_{>0})^r$. We call this point the positive critical point of $W$.

Theorems & Definitions (125)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Generalised Puiseux series Markwig:genPuiseux
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4: The tropicalisation of the torus $T$
  • Definition 2.5: The map $\operatorname{Val_T}$ and identifying $\operatorname{Trop}(T)$ with $N_\mathbb R$
  • Definition 2.6: The tropicalisation of a positive rational map
  • Example 2.7
  • Definition 2.8: $u\mapsto e^u$
  • ...and 115 more