Rational Points on Certain Hyperelliptic Curves over Finite Fields

DOI10.4064/BA55-2-1zbMATH Open1131.11039arXiv0706.1448OpenAlexW2963396100MaRDI QIDQ5293302

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Publication date: 29 June 2007

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Abstract: Let $K$ be a field, $a,binK$ and $abeq0$. Let us consider the polynomials $g_1(x)=x^n+ax+b,g_2(x)=x^n+a{x}^2+bx$, where $n$ is a fixed positive integer. In this paper we show that for each $kgeq2$ the hypersurface given by the equation �egin{equation*} S_{k}^{i}: u^2=prod_{j=1}^{k}g_{i}(x_{j}),quad i=1, 2. end{equation*} contains a rational curve. Using the above and Woestijne's recent results cite{Woe} we show how one can construct a rational point different from the point at infinity on the curves $C_i:y^2=g_i(x),(i=1,2)$ defined over a finite field, in polynomial time.

Full work available at URL: https://arxiv.org/abs/0706.1448

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