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Seminars and Colloquiums
for the week of November 23, 2015


Herivelto Borges, University of Sao Paulo, Monday

TEA TIME - cancelled
3:00 – 3:30 pm, Ayres 401
Monday, Tuesday, & Wednesday

Monday, November 23rd

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: $\lceil \frac{q}{\deg f}\rceil$.

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last updated: May 2018

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