arXiv subjects

Emrah Sercan Yılmaz

Publications and source records attributed to Emrah Sercan Yılmaz.

7 recordsLinked to original sources

arXiv2020

The number of irreducible polynomials over finite fields with vanishing trace and reciprocal trace

We present the formula for the number of monic irreducible polynomials of degree $n$ over the finite field $\mathbb F_q$ where the coefficients of $x^{n-1}$ and $x$ vanish for $n\ge3$. In particular, we give a relation between rational points of algebraic curves over finite fields and the number of elements $a\in\mathbb F_{q^n}$ for which Trace$(a)=0$ and Trace$(a^{-1})=0$. Besides, we apply the formula to give an upper bound on the number of distinct constructions of a family of sequences with good family complexity and cross-correlation measure.

math.NT↗

arXiv2018

Number of Rational points of the Generalized Hermitian Curves over $\mathbb F_{p^n}$

In this paper we consider the curves $H_{k,t}^{(p)} : y^{p^k}+y=x^{p^{kt}+1}$ over $\mathbb F_p$ and and find an exact formula for the number of $\mathbb F_{p^n}$-rational points on $H_{k,t}^{(p)}$ for all integers $n\ge 1$. We also give the condition when the $L$-polynomial of a Hermitian curve divides the $L$-polynomial of another over $\mathbb F_p$.

math.AG↗

arXiv2018

Explicit Maximal and Minimal Curves of Artin-Schreier Type from Quadratic Forms

In this work we present explicit examples of maximal and minimal curves over finite fields in odd characteristic. The curves are of Artin-Schreier type and the construction is closely related to quadratic forms from $\mathbb{F}_{q^n}$ to $\mathbb{F}_q$.

math.AG↗

arXiv2018

On the Zeta Functions of Supersingular Curves

In general, the L-polynomial of a curve of genus $g$ is determined by $g$ coefficients. We show that the L-polynomial of a supersingular curve of genus $g$ is determined by fewer than $g$ coefficients.

math.AG↗

arXiv2018

Divisibility of L-Polynomials for a Family of Artin-Schreier Curves

In this paper we consider the curves $C_k^{(p,a)} : y^p-y=x^{p^k+1}+ax$ defined over $\mathbb F_p$ and give a positive answer to a conjecture about a divisibility condition on $L$-polynomials of the curves $C_k^{(p,a)}$. Our proof involves finding an exact formula for the number of $\mathbb F_{p^n}$-rational points on $C_k^{(p,a)}$ for all $n$, and uses a result we proved elsewhere about the number of rational points on supersingular curves.

math.AG↗

arXiv2018

Slope Estimates for Generalized Artin-Schreier Curves

We provide first slope estimates of the Newton polygon of generalized Artin-Schreier curves, which are proved using the action of Frobenius and Verschiebung on cohomology. We provide a number of applications such as an improved Hasse-Weil bound for this class of curves.

math.AG↗

arXiv2017

The Number of Irreducible Polynomials with the First Two Coefficients Fixed over Finite Fields of Odd Characteristic

In this paper we give a different proof of Kuz'min's result on the number of irreducible polynomials with the first two coefficients fixed. Our technique is to relate the question to the number of points on a curve, and to calculate the L-polynomial of the curve.

math.AG↗