[Turkmath:1318] DEÜ Matematik Bölümü Seminer İlanı -- Leyla Işık

[Celal Cem Sarioglu]

Değerli Liste üyeleri,

Dokuz Eylül Üniversitesi Matematik Bölüm Seminerleri kapsamında, 24 Mayıs
2016 Salı günü saat 11:00'de Leyla Işık (Salzburg University) konuşma

yapacaktır. Konuşma ile ilgili detaylar aşağıdaki gibidir.


---------------------------------------------
Speaker: Leyla Işık

Title: On Complete Maps and Value Sets of Polynomials Over Finite Fields

Abstract:
The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990,
states that for any $d\ge 2$ and any prime $p>(d^2-3d+4)^2$ there is no
complete mapping polynomial in
$\mathbb{F}_p[x]$ of degree $d$.

For arbitrary finite fields $\mathbb{F}_q$, we give a similar result in
terms of the Carlitz rank of a permutation polynomial  rather than its
degree. We prove that if $n<\lfloor q/2\rfloor$, then  there is no complete
mapping in $\mathbb{F}_q [x]$ of Carlitz rank $n$ of small linearity. We
also determine how far permutation polynomials $f$ of Carlitz rank
$n<\lfloor q/2\rfloor$ are from being complete, by studying value sets of
$f(x)+x$.  We provide examples of complete mappings if $n=\lfloor
q/2\rfloor$, which shows that the above bound  cannot be improved in
general.

In this talk, we will also present a new method for constructing complete
mappings of finite fields.  We give a sufficient condition for the
construction of a family of complete mappings of Carlitz rank at most $n$.
Moreover, for $n=4,5,6$ we obtain an explicit construction of complete
mappings.

Finally, we discuss value sets of particular classes of polynomials over
finite fields. We consider a class $\mathcal{F}_{q,n}$ of polynomials of
the form $F(x)=f(x)+x$, where $f$ is a permutation polynomial of Carlitz
rank at most $n$. The study of the spectrum of $\mathcal{F}_{q,n}$ enables
us to obtain a simple description of polynomials $F \in \mathcal{F}_{q,n}$
with
prescribed the value set $V_F$, especially  those avoiding a given set,
like cosets of subgroups of the multiplicative group $\mathbb{F}_q^*$.  The
value set count for such $F$ can also be determined. This yields
 polynomials with evenly distributed values, which have small maximum
count.



Date and Time: 24.05.2016, at 11:00

Place: B206 (DEÜ Mathematics Department)
---------------------------------------------



Saygılarımla,
--
Celal Cem Sarıoğlu
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