Minimal representations of tropical rational functions

Tropical geometry sheds new light on analyzing classical statistical models of piecewise-linear nature.Representing a piecewise-linear function as a tropical rational function, many nontrivial results can be obtained.This paper studies the minimal ones of the above representation.We give two natural notions of complexity for tropical rational functions, the monomial complexity and the factorization complexity.We show that in dimension one, both notions coincide, but this is not true in higher dimensions.We give a canonical representation that is minimal for conewise linear functions on R 2 , which ties to the question of finding canonical representatives for virtual polytopes.We also give comparison bounds between the two notions of complexity.As a proof step, we obtain counting formulas and lower bounds for the number of regions in an arrangement of tropical hypersurfaces, giving a small extension for a result by Montúfar, Ren and Zhang.It also gives a lower bound on the number of vertices in a regular mixed subdivision of a Minkowski sum, giving a small extension for Adiprasito's Lower Bound Theorem for Minkowski sums.We also show that any piecewise-linear function is a linear combination of conewise linear function, which may have implications for model choice for multivariate linear spline regression.