• Title, Summary, Keyword: quartic map

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ON THE GENERAL SOLUTION OF A QUARTIC FUNCTIONAL EQUATION

  • Chung, Jukang-K.;Sahoo, Prasanna, K.
    • Bulletin of the Korean Mathematical Society
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    • v.40 no.4
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    • pp.565-576
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    • 2003
  • In this paper, we determine the general solution of the quartic equation f(x+2y)+f(x-2y)+6f(x) = 4[f(x+y)+f(x-y)+6f(y)] for all x, $y\;\in\;\mathbb{R}$ without assuming any regularity conditions on the unknown function f. The method used for solving this quartic functional equation is elementary but exploits an important result due to M. Hosszu [3]. The solution of this functional equation is also determined in certain commutative groups using two important results due to L. Szekelyhidi [5].

A METHOD OF COMPUTING THE CONSTANT FIELD OBSTRUCTION TO THE HASSE PRINCIPLE FOR THE BRAUER GROUPS OF GENUS ONE CURVES

  • Han, Ilseop
    • Journal of the Korean Mathematical Society
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    • v.53 no.6
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    • pp.1431-1443
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    • 2016
  • Let k be a global field of characteristic unequal to two. Let $C:y^2=f(x)$ be a nonsingular projective curve over k, where f(x) is a quartic polynomial over k with nonzero discriminant, and K = k(C) be the function field of C. For each prime spot p on k, let ${\hat{k}}_p$ denote the corresponding completion of k and ${\hat{k}}_p(C)$ the function field of $C{\times}_k{\hat{k}}_p$. Consider the map $$h:Br(K){\rightarrow}{\prod\limits_{\mathfrak{p}}}Br({\hat{k}}_p(C))$$, where p ranges over all the prime spots of k. In this paper, we explicitly describe all the constant classes (coming from Br(k)) lying in the kernel of the map h, which is an obstruction to the Hasse principle for the Brauer groups of the curve. The kernel of h can be expressed in terms of quaternion algebras with their prime spots. We also provide specific examples over ${\mathbb{Q}}$, the rationals, for this kernel.