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Osculating Geometry and Higher-Order Distance Loci
Sandra Di Rocco, Kemal Rose, Luca Sodomaco
https://arxiv.org/abs/2507.02823 https://

Osculating Geometry and Higher-Order Distance Loci
We discuss the problem of optimizing the distance function from a given point, subject to polynomial constraints. A key algebraic invariant that governs its complexity is the Euclidean distance degree, which pertains to first-order tangency. We focus on the data locus of points possessing at least one critical point of the distance function that is normal to a higher-order osculating space. We propose a novel definition of higher-order distance degree as an intersection-theoretic invariant invo…