Polynomial bound for the number of rational points on curves
Let \(k\) be a number field, with discriminant \(\Delta\). Let \(f\in k[X,Y]\) be a polynomial of degree \(D\). The main theorem of the paper states that the set of affine \(k\)-rational points of \(f(x,y)=0\) is either infinite or bounded by NEWLINE\[NEWLINE \text{exp}\left(2^{3^{D^2}}[K:{\mathbb Q}]^4 \max (h_\infty(F), \log|\Delta|, 1)\right), NEWLINE\]NEWLINE where \(h_\infty(F)\) denotes the logarithm of the maximum archimedean absolute value of the coefficients of \(F\).NEWLINENEWLINEThe paper uses a simple estimate of the number of points of \({\mathbb P}^n(\bar{\mathbb Q})\) of degree \(\leq D\) and of height \(\leq H\) and compares with a previous paper of the same author [Proc. Lond. Math. Soc. (3) 101, No. 3, 759--794 (2010; Zbl 1210.11073)], improving the term of the estimate concerning the height but losing (substantially) on the dependence on the degree.
scientific article; zbMATH DE number 6075042