# Tropical Newton-Puiseux polynomials

## TL;DR

This paper introduces tropical Newton-Puiseux polynomials with rational exponents, providing algorithms for resolving tropical hypersurfaces and studying the complexity of tropical prevarieties, extending tropical algebraic geometry.

## Contribution

It defines tropical Newton-Puiseux polynomials and rational functions, and proves that any tropical polynomial can be resolved using these functions, analogous to algebraic closedness.

## Key findings

- Developed a polynomial complexity algorithm for resolving tropical curves.
- Studied the complexity of resolving tropical prevarieties of arbitrary codimensions.
- Proved that tropical polynomials can be resolved in tropical Newton-Puiseux rational functions.

## Abstract

We introduce tropical Newton-Puiseux polynomials admitting rational exponents. A resolution of a tropical hypersurface is defined by means of a tropical Newton-Puiseux polynomial. A polynomial complexity algorithm for resolubility of a tropical curve is designed. The complexity of resolubility of tropical prevarieties of arbitrary codimensions is studied. Tropical Newton-Puiseux rational functions are introduced, and we prove that any tropical polynomial has a resolution in tropical Newton-Puiseux rational functions (this can be treated as a tropical analog of the algebraic closedness of the field of Newton-Puiseux series).