少系数Thue不等式
Thue inequalities with few coefficients
摘要：
设$F(x, y)$是具有整数系数、次数为$n\geq 3$且在有理数上不可约的二元形式.假设$F$的$n + 1$系数中只有$s + 1$非零。证明了Thue不等式$|F(x,y)|\leq m$有$\ll sm^{2/n}$解，只要$F$的判别式$D(F)$的绝对值足够大.我们还给出了$|F(x,y)|\leq m$的解的个数的一个新的上界，对$F$的判别式没有限制，它主要依赖于$s$和$m$，而对$n$有轻微的依赖.当$m<|D(F)|^{2/(5(n-1))}$时，我们的界变得与$m$无关;当$|D(F)|$足够大时，我们的界也与$n$无关.
英文摘要：
Let $F(x, y)$ be a binary form with integer coefficients, degree $n\geq 3$ and irreducible over the rationals. Suppose that only $s + 1$ of the $n + 1$ coefficients of $F$ are nonzero. We show that the Thue inequality $|F(x,y)|\leq m$ has $\ll sm^{2/n}$ solutions provided that the absolute value of the discriminant $D(F)$ of $F$ is large enough. We also give a new upper bound for the number of solutions of $|F(x,y)|\leq m$, with no restriction on the discriminant of $F$ that depends mainly on $s$ and $m$, and slightly on $n$. Our bound becomes independent of $m$ when $m<|D(F)|^{2/(5(n-1))}$, and also independent of $n$ if $|D(F)|$ is large enough.
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