Quadratically Enriched Tropical Intersections

TL;DR

This paper extends tropical geometry techniques to enriched enumerative geometry over arbitrary fields by proving Bézout's and Bernstein-Kushnirenko theorems within the Grothendieck-Witt ring, connecting combinatorial and algebraic methods.

Contribution

It introduces tropical methods into $ ext{A}^1$-enumerative geometry, enriching classical results with algebraic structures over arbitrary fields.

Findings

Proves Bézout's theorem in the $ ext{GW}(k)$-enriched tropical setting.

Generalizes Bernstein-Kushnirenko theorem for tropical hypersurfaces.

Establishes a link between tropical geometry and $ ext{A}^1$-enumerative geometry.

Abstract

Using tropical geometry one can translate problems in enumerative geometry to combinatorial problems. Thus tropical geometry is a powerful tool in enumerative geometry over the complex and real numbers. Results from $\mathbb{A}^1$-homotopy theory allow to enrich classical enumerative geometry questions and get answers over an arbitrary field. In the resulting area, $\mathbb{A}^1$-enumerative geometry, the answer to these questions lives in the Grothendieck-Witt ring of the base field $k$. In this paper, we use tropical methods in this enriched set up by showing B\'ezout's theorem and a generalization, namely the Bernstein-Kushnirenko theorem, for tropical hypersurfaces enriched in $\operatorname{GW}(k)$.