有限生成域上判别式方程的有效结果
Effective results for discriminant equations over finitely generated domains
AI总结：
本文将关于有限生成整环上判别式方程有限解及有效求解的结果，从整闭整环推广至非整闭整环。
摘要：
设 $A$ 为一个整环，其商域 $K$ 的特征为 $0$，且作为 $\mathbb{Z}$-代数是有限生成的。记 $D(F)$ 为多项式 $F\in A[X]$ 的判别式。进一步，给定一个有限 etale 代数 $Ω$，我们记 $D_{Ω/K}(α)$ 为 $α$ 在 $K$ 上的判别式。对于非零 $δ\in A$，我们考虑方程 \\[ D(F)=δ\\] 在首一多项式 $F\in A[X]$ 中的求解，其中 $F$ 的次数为给定的 $n\geq 2$，且其根位于 $K$ 的给定有限扩域 $G$ 中；以及方程 \\[ D_{Ω/K}(α)=δ\\,\\,\mbox{ in } α\in O, \\] 其中 $O$ 是 $Ω$ 的一个 $A$-序，即 $A$ 在 $Ω$ 中的整闭包的一个子环，它包含 $A$ 以及 $Ω$ 的一组 $K$-基。在我们的书《Discriminant Equations in Diophantine Number Theory》（将由 Cambridge University Press 出版）中，我们证明了：如果 $A$ 在明确定义的意义下被有效给出且是整闭的，那么在自然等价的意义下，上述方程仅有有限多个解，并且可以有效地确定等价类的一个完整代表系。在本文中，我们将这些结果推广到不一定是整闭的整环 $A$。
英文摘要：
Let $A$ be an integral domain with quotient field $K$ of characteristic $0$ that is finitely generated as a $\mathbb{Z}$-algebra. Denote by $D(F)$ the discriminant of a polynomial $F\in A[X]$. Further, given a finite etale algebra $Ω$, we denote by $D_{Ω/K}(α)$ the discriminant of $α$ over $K$. For non-zero $δ\in A$, we consider equations \[ D(F)=δ\] to be solved in monic polynomials $F\in A[X]$ of given degree $n\geq 2$ having their zeros in a given finite extension field $G$ of $K$, and \[ D_{Ω/K}(α)=δ\,\,\mbox{ in } α\in O, \] where $O$ is an $A$-order of $Ω$, i.e., a subring of the integral closure of $A$ in $Ω$ that contains $A$ as well as a $K$-basis of $Ω$. In our book ``Discriminant Equations in Diophantine Number Theory, which will be published by Cambridge University Press we proved that if $A$ is effectively given in a well-defined sense and integrally closed, then up to natural notions of equivalence the above equations have only finitely many solutions, and that moreover, a full system of representatives for the equivalence classes can be determined effectively. In the present paper, we extend these results to integral domains $A$ that are not necessarily integrally closed.
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