The tropical critical point and mirror symmetry

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.