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K-State Today

January 28, 2016

Mathematics Colloquium lecture Jan. 28

By Reta McDermott

Herivelto Borges Filho, University of Sao Paulo, will present "Frobenius Nonclassical Curves and Minimal Value Set Polynomials" as part of the Mathematics Department Colloquium Lecture series at 2:30 p.m. Thursday, Jan. 28, in 122 Cardwell Hall.

The abstract for the lecture is: An irreducible plane curve C defined over a finite field F_q is called Frobenius nonclassical if the image Fr(P) of each simple point P∈C under the Frobenius map lies on the tangent line at P. Otherwise, C is called Frobenius classical. In the latter case, if C has degree d and N as its number of F_q-rational points, then the St\"ohr-Voloch theorem gives N≤d(d+q−1)/2.

Thus if we are able to identify the Frobenius nonclassical curves, we will be left with the remaining curves for which a nice upper bound holds. At the same time, the set of Frobenius nonclassical curves provides a potential source of curves with many rational points.

Filho will discuss the rudiments of the St\"ohr-Voloch theory and present a characterization of the Frobenius non-classical curves of type f(x)=g(y). In particular, we will see that such curves are closely related to the so-called minimal value set polynomials, that is, non-constant polynomials f∈F_q[x] for which V_f:={f(α):α∈F_q} has the minimum possible size:⌈q/deg f⌉.

References: [1] H. Borges, Frobenius nonclassical components of curves with separated variables. Journal of Number Theory, 159 (2016).
[2] St\"ohr, K-O. and Voloch, J.F., Weierstrass Points and Curves over Finite Fields, Proc. London Math. Soc.(3) 52 (1986)1–19.