Counting plane cubic curves over finite fields with a prescribed number of rational intersection points
For each integer k∈[0, 9], we count the number of plane cubic curves defined over a finite
field F q that do not share a common component and intersect in exactly k 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 …
field F q that do not share a common component and intersect in exactly k 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 …
Abstract
For each integer , we count the number of plane cubic curves defined over a finite field that do not share a common component and intersect in exactly -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.
Springer
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