**Seminars and Colloquiums**

for the week of November 23, 2015

for the week of November 23, 2015

*SPEAKER:*

Herivelto Borges, University of Sao Paulo, Monday

*TEA TIME - cancelled*

3:00 – 3:30 pm, Ayres 401

Monday, Tuesday, & Wednesday

3:00 – 3:30 pm, Ayres 401

Monday, Tuesday, & Wednesday

**Monday, November 23rd **

ALGEBRA SEMINAR

TITLE: Frobenius nonclassical curves and minimal value set polynomials

SPEAKER: Herivelto Borges, University of Sao Paulo

TIME: 3:35-4:25

ROOM: Ayres 114

An irreducible plane curve $\mathcal{C}$ defined over a finite field $\mathbf{F}_q$ is called Frobenius nonclassical if the image $Fr(P)$ of each simple point $P \in \mathcal{C}$ under the Frobenius map lies on the tangent line at $P$. Otherwise, $\mathcal{C}$ is called Frobenius classical.

In the latter case, if $\mathcal{C}$ has degree $d$ and $N$ is its number of $\mathbf{F}_q$-rational points, then the St\"ohr-Voloch theorem gives

$$N \leq d(d+q-1)/2.$$

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.

In this talk, I 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\in \mathbf{F}_q[x]$ for which

$$V_f:=\{f(\alpha): \alpha \in \mathbf{F}_q\}$$ has the minimum possible size: $\lceil \frac{q}{\deg f}\rceil$.

*If you are interested in giving or arranging a talk for one of our seminars or colloquiums, please review our calendar. *

*If you have questions, or a date you would like to confirm, please contact colloquium AT math DOT utk DOT edu *