Refined Tropicalizations for Sch\"on Subvarieties of Tori

We introduce a relative refined $\chi_y$-genus for sch\"on subvarieties of algebraic tori. These are rational functions of degree minus the codimension with coefficients in the ring of lattice polytopes. We prove that the relative refined $\chi_y$ turns sufficiently generic intersections into products, and that we can recover the ordinary $\chi_y$-genus by counting lattice points. Applying the tropical Chern character to the relative refined $\chi_y$-genus we obtain a refined tropicalization which is a tropical cycle having rational functions with $\mathbb Q$-coefficients as weights. We prove that the top-dimensional component of the refined tropicalization specializes to the unrefined tropicalization up to sign when setting $y=0$ and show that we can recover the $\chi_y$-genus by integrating the refined tropicalization with respect to a Todd measure.