论文标题
绝对绑定了thue和thue-mahler类型的某些二芬太汀方程的解决方案数量
Absolute Bound On the Number of Solutions of Certain Diophantine Equations of Thue and Thue-Mahler Type
论文作者
论文摘要
令$ f \ in \ mathbb z [x，y] $为不可约的二进制形式$ d \ geq 7 $和content One。令$α$为$ f（x，1）$的根，并假设字段扩展$ \ mathbb q（α）/\ mathbb q $是Galois。我们证明，对于每一个足够大的Prime Power $ p^k $，thue thue type type $$ $ f（x，y）的diophantine方程的解决方案| = tp^k $$ in Integers $（x，y，t）$，使得$ \ gcd（x，y）= 1 $和$ 1 \ leq t \ leq（p^k）^λ$不超过$ 24 $。在这里，$λ=λ（d）$是某种积极的，单调的功能，以$ d $趋向于无穷大。我们还证明，对于每个足够大的质数$ p $，thue-mahler type type $$ | f（x，y）diophantine方程的解决方案数量（x，y）| = tp^z $$ in Integers $（x，y，z，t）$，使得$ \ gcd（x，y）= 1 $，$ z \ geq 1 $和$ 1 \ leq t \ leq t \ leq（p^z）^{\ frac {\ frac {10d -61}} {20d + 40}} {20d + 40}} $不超过1992年的组合。近似，即广义的非架构间隙原则和thue-siegel原理。
Let $F \in \mathbb Z[x, y]$ be an irreducible binary form of degree $d \geq 7$ and content one. Let $α$ be a root of $F(x, 1)$ and assume that the field extension $\mathbb Q(α)/\mathbb Q$ is Galois. We prove that, for every sufficiently large prime power $p^k$, the number of solutions to the Diophantine equation of Thue type $$ |F(x, y)| = tp^k $$ in integers $(x, y, t)$ such that $\gcd(x, y) = 1$ and $1 \leq t \leq (p^k)^λ$ does not exceed $24$. Here $λ= λ(d)$ is a certain positive, monotonously increasing function that approaches one as $d$ tends to infinity. We also prove that, for every sufficiently large prime number $p$, the number of solutions to the Diophantine equation of Thue-Mahler type $$ |F(x, y)| = tp^z $$ in integers $(x, y, z, t)$ such that $\gcd(x, y) = 1$, $z \geq 1$ and $1 \leq t \leq (p^z)^{\frac{10d - 61}{20d + 40}}$ does not exceed 1992. Our proofs follow from the combination of two principles of Diophantine approximation, namely the generalized non-Archimedean gap principle and the Thue-Siegel principle.
