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arXiv · 2605.24821

Counting Rational Points on Danielewski and Double Danielewski Surfaces over Finite Fields

Sakshi Gupta·Anit Kuckian·Indranath Sengupta
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Abstract

Let $\Fq$ be the finite field with $q$ elements. We study the number of $\Fq$-rational points on Danielewski and double Danielewski surfaces. For Danielewski surfaces, the point count is reduced to the number of roots of $P(Z)$ over $\Fq.$ For double Danielewski surfaces, one has to count the number of tuples $(\be,\g)\in\Fq^2$, such that $P(0,\g)=0$, $Q(0,\be,\g)=0$ hold simultaneously. We compute these numbers using gcd methods, resultants, character sums, Gauss sums, and the K\"onig--Rados theorem. We obtain explicit formulas in several structured cases, derive general bounds, and give a Macaulay2 algorithm for verification and show an intresting connection between the number of $\Fq$-rational points of these surfaces and polygonal numbers.

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Sakshi Gupta, Anit Kuckian, Indranath Sengupta. 2026-05-24. Counting Rational Points on Danielewski and Double Danielewski Surfaces over Finite Fields. https://arxiv.org/abs/2605.24821

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