论文标题
系数很少的不平等现象
Thue inequalities with few coefficients
论文作者
论文摘要
令$ f（x，y）$为具有整数系数的二进制形式，$ n \ geq 3 $且在理性方面不可约。假设$ n + 1 $系数的$ f $的$ s + 1 $是非零的。我们表明，不平等$ | f（x，y）| \ leq m $具有$ \ ll sm^{2/n} $解决方案，规定$ f $的判别$ d（f）$的绝对值就足够大。我们还为$ | f（x，y）| \ leq m $的解决方案的数量提供了新的上限，对$ f $的判别物没有限制，这主要取决于$ s $和$ m $，而略微限制在$ n $上。当$ m <| d（f）|^{2/（5（n-1））} $时，我们的界限将独立于$ m $，并且如果$ | d（f）| $足够大，也独立于$ n $。
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.
