Counting Rational Points on Danielewski and Double Danielewski Surfaces over Finite Fields
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arXiv
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| Format: | Preprint |
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2026
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| author | Gupta, Sakshi Kuckian, Anit Sengupta, Indranath |
| author_facet | Gupta, Sakshi Kuckian, Anit Sengupta, Indranath |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_24821 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Counting Rational Points on Danielewski and Double Danielewski Surfaces over Finite Fields Gupta, Sakshi Kuckian, Anit Sengupta, Indranath Number Theory Commutative Algebra 11T06, 11T99 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. |
| title | Counting Rational Points on Danielewski and Double Danielewski Surfaces over Finite Fields |
| topic | Number Theory Commutative Algebra 11T06, 11T99 |
| url | https://arxiv.org/abs/2605.24821 |