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Article Citation - WoS: 3Citation - Scopus: 4Some Permutations and Complete Permutation Polynomials Over Finite Fields(Tubitak Scientific & Technological Research Council Turkey, 2019) Ongan, Pınar; Temür, Burcu Gülmez; Gülmez Temür, BurcuIn this paper we determine $b\\in F_{q^n}^\\ast$ for which the polynomial $f(x)=x^{s+1}+bx\\in F_{q^n}\\left[x\\right]$ is a permutationpolynomial and determine $b\\in F_{q^n}^\\ast$ for which the polynominal $f(x)=x^{s+1}+bx\\in F_{q^n}\\left[x\\right]$ is a complete permutationpolynomial where $s=\\frac{q^n-1}t,\\;t\\in\\mathbb{Z}^+$ such that $\\left.t\\;\\right|\\;q^n-1$.Article Citation - WoS: 1An exhaustive computer search for finding new curves with many points among fibre products of two Kummer covers over F5 and F7(Tubitak Scientific & Technological Research Council Turkey, 2013) Ozbudak, Ferruh; Temur, Burcu Gulmez; Yayla, Oguz; Gülmez Temür, BurcuIn this paper we make an exhaustive computer search for finding new curves with many points among fibre products of 2 Kummer covers of the projective line over F-5 and F-7. At the end of the search, we have 12 records and 6 new entries for the current Table of Curves with Many Points. In particular, we observe that the fibre product y(1)(3)= 5(x + 2)(x + 5)/x, y(2)(3)= 3x(2()x + 5)/x + 3 over F-7 has genus 7 with 36 rational points. As this coincides with the Ihara bound, we conclude that the maximum number N-7(7) of F-7-rational points among all curves of genus 7 is 36. Our exhaustive search has been possible because of the methods given in the recent work by Ozbudak and Temur (2012) for determining the number of rational points of such curves.Article Citation - WoS: 1Citation - Scopus: 1On Some Permutation Trinomials in Characteristic Three(Hacettepe Univ, Fac Sci, 2025) Temür, Burcu Gülmez; Özkaya, Buket; Gülmez Temür, BurcuIn this paper, we determine the permutation properties of the polynomial x3 +xq+2 −x4q−1 over the finite field Fq2 in characteristic three. Moreover, we consider the trinomials of the form x4q−1 + x2q+1 ± x3. In particular, we first show that x3 + xq+2 − x4q−1 permutes Fq2 with q = 3m if and only if m is odd. This enables us to show that the sufficient condition in [34, Theorem 4] is also necessary. Next, we prove that x4q−1 + x2q+1 − x3 permutes Fq2 with q = 3m if and only if m ̸≡ 0 (mod 4). Consequently, we prove that the sufficient condition in [20, Theorem 3.2] is also necessary. Finally, we investigate the trinomial x4q−1 + x2q+1 + x3 and show that it is never a permutation polynomial of Fq2 in any characteristic. All the polynomials considered in this work are not quasi-multiplicative equivalent to any known class of permutation trinomials.
