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 König--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.
| Subjects: | Number Theory (math.NT); Commutative Algebra (math.AC) |
| MSC classes: | 11T06, 11T99 |
| Cite as: | arXiv:2605.24821 [math.NT] |
| (or arXiv:2605.24821v1 [math.NT] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24821 arXiv-issued DOI via DataCite (pending registration) |
From: Indranath Sengupta [view email]
[v1]
Sun, 24 May 2026 02:23:47 UTC (26 KB)
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