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Height coincidences in products of the projective line

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We consider hypersurfaces in \({\mathbb {P}}_1^n\) that contain a generic sequence of small dynamical height with respect to a split map and project onto \(n-1\) coordinates. We show that these hypersurfaces satisfy strong coincidence relations between their points with zero height coordinates. More precisely, it holds that in a Zariski-open dense subset of such a hypersurface \(n-1\) coordinates have height zero if and only if all coordinates have height zero. This is a key step in the resolution of the dynamical Bogomolov conjecture for split maps.

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We thank the referee for their careful reading of the article and for their comments and suggestions. We also thank the department of mathematics in Basel and Philipp Habegger for enlightening discussions. N.M.M. acknowledges the support from NSF grant DMS-2200981. R.W. acknowledges support from the SNF grant “Diophantine Equations: Special Points, Integrality, and Beyond” (n\(^\circ \) 200020_184623).

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Correspondence to Niki Myrto Mavraki.

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Mavraki, N.M., Schmidt, H. & Wilms, R. Height coincidences in products of the projective line. Math. Z. 304, 26 (2023).

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