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arXiv · 2003.13944

Counting Plane Cubic Curves over Finite Fields with a Prescribed Number of Rational Intersection Points

Nathan Kaplan·Vlad Matei
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Abstract

For each integer $k \in [0,9]$, we count the number of plane cubic curves defined over a finite field $\mathbb{F}_q$ that do not share a common component and intersect in exactly $k\ \mathbb{F}_q$-rational points. We set this up as a problem about a weight enumerator of a certain projective Reed-Muller code. The main inputs to the proof include counting pairs of cubic curves that do share a common component, counting configurations of points that fail to impose independent conditions on cubics, and a variation of the MacWilliams theorem from coding theory.

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Nathan Kaplan, Vlad Matei. 2020-03-31. Counting Plane Cubic Curves over Finite Fields with a Prescribed Number of Rational Intersection Points. https://doi.org/10.1007/s40879-021-00472-x

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