Polar degrees and closest points in codimension two

TL;DR

This paper provides a combinatorial method to compute Euclidean distance and polar degrees of codimension two toric varieties using Gale dual matrices, enabling efficient analysis of larger examples.

Contribution

It introduces a new combinatorial approach to calculate invariants of codimension two toric varieties from Gale dual matrices, simplifying and speeding up computations.

Findings

Efficient formulas for Euclidean distance degree and polar degrees

Applicable to larger, more complex examples

Simplifies previous algebraic and geometric methods

Abstract

Suppose that $X_A\subset \mathbb{P}^{n-1}$ is a toric variety of codimension two defined by an $(n-2)\times n$ integer matrix $A$, and let $B$ be a Gale dual of $A$. In this paper we compute the Euclidean distance degree and polar degrees of $X_A$ (along with other associated invariants) combinatorially working from the matrix $B$. Our approach allows for the consideration of examples that would be impractical using algebraic or geometric methods. It also yields considerably simpler computational formulas for these invariants, allowing much larger examples to be computed much more quickly than the analogous combinatorial methods using the matrix $A$ in the codimension two case.