Degenerations of Univariate Sparse Polynomials

The work is devoted to the study of degenerations of complex polynomials of one variable by methods of tropical geometry. This kind of study begins with a standard question: to what extent does the tropicalization of such a degeneracy (i.e., the central layer of a family of polynomials) store information about the geometry and topology of a general polynomial in the family? In this paper, we study the topological version of this question. As is known, the complex polynomial of one variable defines the branched covering of the Riemann sphere by the Riemann sphere. For such coverings, a T-equivalence relation can be introduced, considering the coverings which may be obtained from each other by the homeomorphisms of the image and the preimage to be equivalent. In this paper, we review the available results in this and related fields, and also present an approach that will probably help answer the general question of T-classification of polynomials with a given support. This approach uses tropical geometry, namely, we consider a one-dimensional family of polynomials with a given support and branch type and try to determine to what extent the tropicalization of this family preserves information about a general element of this family.