How to Tropicalize a non-Archimedean Lattice

The tropicalization of a linear space over a non-archimedean field is a tropical linear space. In this paper, we present a method for computing the tropicalization of any lattice over a valuation ring. The resulting tropical semimodule is the support of a polyhedral complex constructed from a certain multilinear polynomial we call the entropy polynomial. The key idea in our argument is the tropicalization of Haar measures on lattices over local fields.

💡 Research Summary

This paper establishes a comprehensive theory and computational framework for tropicalizing lattices over non-Archimedean fields. The core problem is extending the well-studied tropicalization of linear spaces to the more general setting of lattices (finitely generated submodules over a valuation ring).

The author introduces two central combinatorial objects associated with a lattice L: the “entropy vector” h(L) and the “entropy polynomial” φ_L(v). The entropy vector records, for each subset J of coordinates, the minimum valuation of the corresponding minors of any matrix generating L. This data is supermodular. The entropy polynomial is a tropical polynomial defined as φ_L(v) = max_{J⊂