奇异情形下的Skolem-Abouzaid定理
The Skolem-Abouzaid theorem in the singular case
摘要：
设F（X;Y）在Q[X;Y]中是Q-不可约多项式.在1929年Skolem证明了以下定理：“假设F（0;0）= 0。则对每个非零整数d，方程F（X;Y）= 0在整数（X;Y）中只有n个解，且gcd（X;Y）= d”。Skolem方法允许人们明确地根据多项式F和整数d的系数来约束解。2008年，Abouzaid对Skolem定理进行了广泛的推广。他延长它在两个方向：首先，他研究的解决方案不仅在合理的整数，但在任意代数数。其次，他不仅用对数gcd来界定解，而且得到了坐标的高度和它们的对数gcd之间的一种渐近关系。不幸的是，Abouzaid假设比Skolem稍微更严格：他不仅假设点（0;0）属于平面曲线F（X;Y）= 0，而且假设（0;0）是该曲线上的非奇点。本文的目的是摆脱这种非奇异性假设。
英文摘要：
Let F(X;Y) in Q[X;Y] be a Q-irreducible polynomial. In 1929 Skolem proved the following theorem: "Assume that F(0;0) = 0. Then for every non-zero integer d, the equation F(X;Y) = 0 has only finitely many solutions in integers (X;Y) with gcd(X;Y) = d". Skolem method allows one to bound the solutions explicitly in terms of the coefficients of the polynomial F and the integer d. In 2008, Abouzaid gave a far-going generalization of Skolem theorem. He extended it in two directions: first, he studied solutions not only in rational integers, but in arbitrary algebraic numbers. Second, he not only bounded the solution in terms of the logarithmic gcd, but obtained a sort of asymptotic relation between the heights of the coordinates and their logarithmic gcd. Unfortunately, Abouzaid assumption is slightly more restrictive than Skolem: he assumes not only that the point (0;0) belongs to the plane curve F(X;Y) = 0, but that (0;0) is a non-singular point on this curve. The purpose of the present article is to get rid of this non singularity hypothesis.
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