Polar degrees and closest points in codimension two

Abstract

Suppose that [Formula: see text] is a toric variety of codimension two defined by an [Formula: see text] integer matrix [Formula: see text], and let [Formula: see text] be a Gale dual of [Formula: see text]. In this paper, we compute the Euclidean distance degree and polar degrees of [Formula: see text] (along with other associated invariants) combinatorially working from the matrix [Formula: see text]. 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 [Formula: see text] in the codimension two case.