有理函数的Hadamard环中的方程
Equations in the Hadamard ring of rational functions
摘要：
设k为数域。众所周知，由k上有理函数的泰勒系数组成的序列集合在分量运算下是闭的，因此它可以具有环结构。一个猜想由于皮索问，如果（扩大领域后）一个可以采取d次根在这个环，提供d次根的系数可以采取在k。这在第二作者的前一篇论文中被证明是正确的;在这篇文章中，我们将这个结果推广到更一般的一般方程，monic在Y中，其中前一种情况可以恢复为g（X，Y）=X^d-Y=0。结合这一点与阿达玛商定理的普尔谢和范德Poorten，我们能够摆脱幺一的限制，并有一个定理，推广了这两个结果。
英文摘要：
Let k be a number field. It is well known that the set of sequences composed by Taylor coefficients of rational functions over k is closed under component-wise operations, and so it can be equipped with a ring structure. A conjecture due to Pisot asks if (after enlarging the field) one can take d-th roots in this ring, provided d-th roots of coefficients can be taken in k. This was proved true in a preceding paper of the second author; in this article we generalize this result to more general equations, monic in Y, where the former case can be recovered for g(X,Y)=X^d-Y=0. Combining this with the Hadamard quotient theorem by Pourchet and Van der Poorten, we are able to get rid of the monic restriction, and have a theorem that generalizes both results.
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