Geometry over the tropical dual numbers

TL;DR

This paper introduces tropical dual numbers as an algebraic extension of the tropical semiring, enabling the use of ideals and recovering Euclidean topology in tropical geometry, thus providing a new algebraic framework.

Contribution

It presents the concept of tropical dual numbers, allowing for ideal-based approaches and Euclidean topology in tropical geometry, which was not possible with previous congruence-based methods.

Findings

Tropical dual numbers extend tropical semiring with ideal-based structure.

They recover Euclidean topology on affine tropical spaces.

They establish a tropical Zariski topology with non-linear loci of ideals.

Abstract

We introduce tropical dual numbers as an extension of tropical semiring. By this innovation, one can work with honest ideals, instead of congruences, and recover the Euclidean topology on affine tropical spaces similar to Zariski's approach in classical algebraic geometry. The bend loci of an ideal over tropical dual numbers coincides with the bend loci of a congruence over tropical semiring and this enables tropical dual numbers to serve as an algebraic structure for tropical geometry. Tropical Zariski topology on affine tropical spaces whose closed sets are non-linear loci of ideals over tropical dual numbers offers an alternative point of view to strong Zariski topology defined by Giansiracusa.