Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4583316 | Finite Fields and Their Applications | 2007 | 5 Pages |
Abstract
Let d(q) denote the minimal degree of a smooth projective plane curve that is defined over the finite field Fq and does not contain Fq rational points. We are interested in the asymptotic behavior of d(q) for qââ. To the best of the author's knowledge the problem of estimating the asymptotic behavior of d(q) was not considered previously. In this note we establish the following bounds:(1)14⩽lim̲qââlogqd(q)⩽13. More specifically, for every characteristic p>3 we construct a sequence of pointless Fermat curvesxdk+ydk+zdk=0,over Fpmk, such that limkââlogpmkdk=1/3.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Sergey Yekhanin,