Thue's equation and a conjecture of Siegel
In his fundamental paper ``Über einige Anwendungen diophantischer Approximationen'' [Abh. Preuss. Akad. Wiss., Phys.-Math. Kl. 1929, No.1, 1--70 (1929; JFM 56.0180.05)] \textit{C. L. Siegel} conjectured that the number of points \(x,y\in {\mathbb{Z}}\) on a curve of positive genus defined by an equation \(f(x,y)=0\) for some polynomial f(x,y)\(\in {\mathbb{Z}}[x,y]\) is bounded above by a number depending only on the number of nonzero coefficients of f. In this form, the conjecture is not true. For Thue's equation \[F(x,y)=h\quad\text{in }x,y\in\mathbb Z, \tag{1}\] where \(F\in {\mathbb{Z}}[x,y]\) is an irreducible binary form of degree \(r\ge 3\), one can state the modified Siegel's conjecture that the number of solutions of (1) has an upper bound depending only on \(h\) and the number \(s+1\) of nonzero coefficients of \(F\). The authors prove that this modified conjecture is true. More generally, let \(Z_F(h)\) be the number of solutions of the inequality \(| F(x,y)| \leq h\) in \(x,y\in\mathbb Z\). Then \[ Z_ F(h)\leq c_1 s^ 2 h^{2/r}(1+\log h^{1/r}), \] where \(c_1\) is some absolute constant. If \(r\geq \max (4s,s \log^ 3s)\) then this upper bound can be improved to \(c_2s^2h^{2/r}.\) The authors also compare \(Z_F(h)\) with the area \(A_F(h)\) of the set \[ \{(\xi,\eta)\in \mathbb R^2: | F(\xi,\eta)| \leq h\}.\] Note that \(A_ F(h)=A_ F(1)h^{2/r}\). The authors prove that if \(r\ge 4s\), then \(A_ F(1)\leq c_3(rs^2)^{2s/r}\) and \[ | Z_F(h)-A_F(1)h^{2/r}| \leq c_4(s)(h^{1/(r-2s)}+h^{1/r} \log h^{1/r}), \] where \(c_4(s)\) depends only on \(s\).
scientific article; zbMATH DE number 4068976
scientific article