Euclidean distance degree defect of singular projective varieties
Abstract
The unit Euclidean distance degree and the generic Euclidean distance degree are two well-studied invariants of projective varieties. These quantities measure the algebraic complexity of nearest-point problems on a variety, and in many examples arising in optimization, engineering, statistics, and data science, there is a significant gap between them. We refer to this difference as the defect of the Euclidean distance (ED) degree. In this paper, we provide a constructible enhancement and a topological formula for the defect of the ED degree of an arbitrary complex projective variety, extending our previous results from the smooth setting. Since the generic Euclidean distance degree is typically more tractable, our approach offers a new method for computing ED degrees in broad generality.
Related
- Hyperbolicity and fundamental groups of complex quasi-projective varieties (III): applications
- Singular solutions from simple Euclidean Jordan algebras
- Automorphism groups and Homogeneous fibrations on projective varieties with nef anticanonical divisors
- Characteristic-free characterizations of projective space for smooth toric varieties
- Rank two cohomology jump loci on projective varieties
BibTeX
@misc{maxim2026euclidean,
title = {{Euclidean distance degree defect of singular projective varieties}},
author = {Laurenţiu G. Maxim and Jose Israel Rodriguez and Botong Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.13726},
eprint = {2605.13726},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.13726},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Euclidean distance degree defect of singular projective varieties
AU - Laurenţiu G. Maxim
AU - Jose Israel Rodriguez
AU - Botong Wang
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2605.13726
UR - https://arxiv.org/abs/2605.13726
AB - The unit Euclidean distance degree and the generic Euclidean distance degree are two well-studied invariants of projective varieties. These quantities measure the algebraic complexity of nearest-point problems on a variety, and in many examples arising in optimization, engineering, statistics, and data science, there is a significant gap between them. We refer to this difference as the defect of the Euclidean distance (ED) degree. In this paper, we provide a constructible enhancement and a topological formula for the defect of the ED degree of an arbitrary complex projective variety, extending our previous results from the smooth setting. Since the generic Euclidean distance degree is typically more tractable, our approach offers a new method for computing ED degrees in broad generality.
ER -