Solymosi--Zahl: Improved Elekes--Szabó type estimates using proximity

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József Solymosi and Joshua Zahl, arXiv:2211.13294 (2022). Full primary paper.

Theorem 1.4 bounds the intersection of an irreducible non-special algebraic surface with a real Cartesian product. For three sets of size at most $Q$, the bound is $O_d(Q^{12/7})$, with constant depending only on the polynomial degree. Axis-parallel cylinders and surfaces with local additive-group structure are the exceptions.

Section 1.1 explicitly applies the theorem to distances from three noncollinear planar anchors, giving at least $cN^{7/12}$ distinct distances to $N$ points. The full theorem and application were checked. This improves the earlier $N^{6/11}$ bound.