A Geometric Criterion for Degeneracy in the Elekes-Szabó Theorem
Abstract
The Elekes-Szabó theorem establishes that an irreducible algebraic hypersurface Z(F) contains few grid points unless it exhibits a specific group-related structure. Identifying this structure from the polynomial F is a challenging problem in combinatorial geometry. Our first main result (Theorem 2.3) provides a local geometric criterion to detect such group-related hypersurfaces. By applying this criterion, we develop a geometric framework for boundary varieties, which are defined by the vanishing of partial derivatives along Z(F). In Theorem 2.4, we show that for group-related varieties, these boundary varieties must be contained in coordinate slices. This gives a strict geometric constraint on the loci where Z(F) becomes tangent to coordinate directions. As an application, we study configurations formed by d coordinate-grid hyperplane families together with a one-parameter polynomial family of hyperspheres in ℝd. If one chooses n members from each of these d+1 families and obtains Ω(nd−η) common incidence points, then the hypersphere family is forced to have a very restricted form: it is concentric in dimensions d ≥ 3, and in dimension $2$ it is either concentric or consists of fixed-radius circles whose centres lie on a line parallel to a coordinate axis. We also generalize the pinned distance problem initiated by Elekes and Szabó for three points in the plane to d+1 points in ℝd. More precisely, in Theorem 2.8 we prove that if d+1 families of hyperspheres centred at fixed points determine Ω(nd−η) points, each lying on one hypersphere from each family, then the centres must be affinely dependent.
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BibTeX
@misc{makhul2026geometric,
title = {{A Geometric Criterion for Degeneracy in the Elekes-Szabó Theorem}},
author = {Mehdi Makhul},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.03366},
eprint = {2607.03366},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.03366},
}
RIS
TY - UNPB
TI - A Geometric Criterion for Degeneracy in the Elekes-Szabó Theorem
AU - Mehdi Makhul
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2607.03366
UR - https://arxiv.org/abs/2607.03366
AB - The Elekes-Szabó theorem establishes that an irreducible algebraic hypersurface Z(F) contains few grid points unless it exhibits a specific group-related structure. Identifying this structure from the polynomial F is a challenging problem in combinatorial geometry. Our first main result (Theorem 2.3) provides a local geometric criterion to detect such group-related hypersurfaces. By applying this criterion, we develop a geometric framework for boundary varieties, which are defined by the vanishing of partial derivatives along Z(F). In Theorem 2.4, we show that for group-related varieties, these boundary varieties must be contained in coordinate slices. This gives a strict geometric constraint on the loci where Z(F) becomes tangent to coordinate directions. As an application, we study configurations formed by d coordinate-grid hyperplane families together with a one-parameter polynomial family of hyperspheres in ℝ^d. If one chooses n members from each of these d+1 families and obtains Ω(n^d−η) common incidence points, then the hypersphere family is forced to have a very restricted form: it is concentric in dimensions d ≥ 3, and in dimension $2$ it is either concentric or consists of fixed-radius circles whose centres lie on a line parallel to a coordinate axis. We also generalize the pinned distance problem initiated by Elekes and Szabó for three points in the plane to d+1 points in ℝ^d. More precisely, in Theorem 2.8 we prove that if d+1 families of hyperspheres centred at fixed points determine Ω(n^d−η) points, each lying on one hypersphere from each family, then the centres must be affinely dependent.
ER -