Sieving for rational points on hyperelliptic curves (Q2723535)

Let \(f(x)\) be a polynomial with integer coefficients of degree \(d=2n, n \geq 2\) without repeated roots. In this paper an algorithm for finding rational points on the hyperelliptic curve \(C: y^2 = f(x)\) is presented. The method is an improved sieving process. NEWLINENEWLINENEWLINELet \(I\) be an interval such that \(f(x)\geq 0\) for all \(x \in I.\) Assume that \((X/Z,Y/Z^n)\in C({\mathbb Q})\) with \(X,Z\) coprime integers such that \(X/Z \in I.\) Then \(Y^2 = F(X,Z) = Z^df(X/Z).\) NEWLINENEWLINENEWLINELet \(\alpha, \beta\) be coprime integers, and \(F(\alpha,\beta)=\gamma \delta^2\), where \(|\gamma|>1\) is square-free. Set \(N=|\gamma|\), if \(\gamma \equiv 1 \pmod 4\) and \(N=4|\gamma|\) otherwise. Further let \(B= \{ p_1^{r_1}\dots p_l^{r_l}q: p_1,\dots,p_l \) the distinct prime divisors of \(N\), \(r_1,\dots,r_l\) bounded non-negative integers, \(q=1\) or a suitable prime\(\}\). NEWLINENEWLINENEWLINEThe main result, is that if \(\zeta \in \{1,-1\}\) is chosen appropriately then there exist \(P\in B\) and \(h\in H\) such that NEWLINE\[NEWLINE\beta X - \alpha Z \equiv \zeta Ph \pmod {PN},NEWLINE\]NEWLINE where \(H\) is a subgroup in \(({\mathbb Z}/N{\mathbb Z})^*.\) NEWLINENEWLINENEWLINEIn the proof quadratic reciprocity is used. NEWLINENEWLINENEWLINEChoosing quadruples \((\alpha, \beta,\delta,\gamma)\) one can generate several systems of congruences for the unknown \(X\) and \(Z\). The author describes and analyses a method for the solution of such systems, too. There are two examples that illustrate the power of the method.