A Geometric Criterion for Degeneracy in the Elekes-Szabó Theorem

The Elekes-Szab\'{o} 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 $\mathbb{R}^d$. If one chooses $n$ members from each of these $d+1$ families and obtains $\Omega(n^{d-\eta})$ common incidence points, then the hypersphere family is forced to have a very restricted form: it is concentric in dimensions $d \geq 3$, 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 Szab\'{o} for three points in the plane to $d+1$ points in $\mathbb{R}^d$. More precisely, in Theorem 2.8 we prove that if $d+1$ families of hyperspheres centred at fixed points determine $\Omega(n^{d-\eta})$ points, each lying on one hypersphere from each family, then the centres must be affinely dependent.