ON THE DISTANCE TO DISCRIMINANTS AND OSCULATING VARIETIES
Abstract
We study the problem of finding the closest singular hypersurface to a given nonsingular one from algebro-geometric point of view. The set of singular hypersurfaces Ξ of a given degree is a projective variety dual to the Veronese variety. However, as was shown by Raffalli [Raf14], the Bombieri-Weyl (also known as apolar) distance from a general real hypersurface to Ξ could be minimized at a singular point of Ξ. We study the singular locus of Ξ and its irreducible component, consisting of hypersurfaces with cuspidal singularities. This variety is projectively dual to the tangential variety of the Veronese variety. We compute its Euclidean Distance Degree using topological tools and deduce a formula for the number of critical points of the distance function to Ξ from a general hypersurface. We also initiate the study of the analogous problem for higher osculating varieties to the Veronese variety. Furthermore, we establish a new extremal property of Chebyshev polynomials (conjectured in [Raf14]) by characterizing real-rooted binary forms that maximize the distance to Ξ.
Related
- Quadratic hulls of osculating varieties of rational normal curves
- Euclidean distance degree defect of singular projective varieties
- Discriminants of Hermitian forms, maximal degenerations and Kontsevich's tropical approach to the Hodge conjecture
- Matrix Completion Discriminants
- Whitney fold and cusp for algebraic surfaces, and singularities of discriminants
BibTeX
@misc{kozhasov2026distance,
title = {{ON THE DISTANCE TO DISCRIMINANTS AND OSCULATING VARIETIES}},
author = {Khazhgali Kozhasov and Bernard Mourrain and Adam Parusiński},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06538},
eprint = {2610.06538},
archivePrefix = {arXiv},
url = {https://inria.hal.science/hal-05777819v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - ON THE DISTANCE TO DISCRIMINANTS AND OSCULATING VARIETIES
AU - Khazhgali Kozhasov
AU - Bernard Mourrain
AU - Adam Parusiński
PY - 2026
DA - 2026/10/05
PB - HAL
DO - 10.48550/arxiv.2610.06538
UR - https://inria.hal.science/hal-05777819v1
AB - We study the problem of finding the closest singular hypersurface to a given nonsingular one from algebro-geometric point of view. The set of singular hypersurfaces Ξ of a given degree is a projective variety dual to the Veronese variety. However, as was shown by Raffalli [Raf14], the Bombieri-Weyl (also known as apolar) distance from a general real hypersurface to Ξ could be minimized at a singular point of Ξ. We study the singular locus of Ξ and its irreducible component, consisting of hypersurfaces with cuspidal singularities. This variety is projectively dual to the tangential variety of the Veronese variety. We compute its Euclidean Distance Degree using topological tools and deduce a formula for the number of critical points of the distance function to Ξ from a general hypersurface. We also initiate the study of the analogous problem for higher osculating varieties to the Veronese variety. Furthermore, we establish a new extremal property of Chebyshev polynomials (conjectured in [Raf14]) by characterizing real-rooted binary forms that maximize the distance to Ξ.
ER -