Rank of tropical curves and tropical hypersurfaces

TL;DR

This paper develops formulas, bounds, and algorithms to compute the dimension of deformation spaces of tropical curves and hypersurfaces, linking tropical geometry with algebraic deformation theory.

Contribution

It introduces new precise formulas, bounds, and algorithms for the deformation space dimensions of tropical curves and hypersurfaces, enhancing computational and theoretical understanding.

Findings

Derived exact formulas for deformation space dimensions

Established bounds for various classes of tropical objects

Provided algorithms for practical computation of these dimensions

Abstract

This paper is devoted to the bounding and computation of the dimension of deformation spaces of tropical curves and hypersurfaces. This characteristic is interesting in light of the fact that it often coincides with the dimension of equisingular (equigeneric etc.) deformation spaces of algebraic curves and hypersurfaces. In this paper, we obtain a series of precise formulas, upper and lower bounds, and algorithms for computing dimension of deformation spaces of various classes of tropical curves and hypersurfaces.