A positive/tropical critical point theorem and mirror symmetry

Call a Laurent polynomial W ‘complete’ if its Newton polytope is full-dimensional with zero in its interior. Suppose W is a Laurent polynomial with coefficients in the positive part of the field of (generalised) Puiseaux series. Here a Puiseaux or generalised Puiseux series (with exponents in $R$) is called ‘positive’ if the coefficient of its leading term is in $R_{>0}$. We show that W has a unique positive critical point $p_{\mathrm{crit}}$, i.e. all of whose coordinates are positive, if and only if W is complete. For any complete, positive Laurent polynomial W in r variables we also obtain from its positive critical point $p_{\mathrm{crit}}$ a canonically associated ‘tropical critical point’ $d_{\mathrm{crit}}\in R^r$ by considering the valuations of the coordinates of $p_{\mathrm{crit}}$. Moreover we give a finite recursive construction of $d_{\mathrm{crit}}$ in terms of a generalisation of the Newton polytope that we call the ‘Newton datum’ of W.

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 show that our theorem carries over to the case where the exponents of monomials appearing in W are not integral but in $R$, even though W is then no longer Laurent.

Finally, we describe applications to both algebraic and symplectic toric geometry inspired by mirror symmetry. On the one hand, in the algebraic context of a complete toric variety $X_{\Sigma }$ we apply our results to obtain for any divisor class $\left[D\right]$ satisfying a certain integrality property, a canonical choice of torus-invariant representative. This generalises the standard toric boundary divisor of $X_{\Sigma }$ to divisor classes other than the anti-canonical class. On the other hand, our result generalises a result of [11] and relates to the construction of canonical non-displaceable Lagrangian tori for toric symplectic orbifolds using [13], [37].