平方在$p$ -adic亚纯函数域上的一元二次多项式表示
Representation of squares by monic second degree polynomials in the field of $p$-adic meromorphic functions
摘要：
本文证明了$p$ -adic亚纯函数域上平方的二次多项式表示的一个结果，从而正解了该域上的$n$平方问题（即，存在一个常数$M$的问题，使得任何序列$(x_n^2)$的$M$ -不是所有常数-其第二个差是常数序列$(2)$的平方对于某些$x$满足$x_n^2=(x+n)^2$）。我们证明了（根据Vojta的作品）一个类似的结果，功能领域的特征零，并根据猜想Besieri，一个类似的结果数域。利用Büchi的一个论证，我们证明了所得结果改进了关于$p$ -adic亚纯函数域和$p$ -adic整函数环的不可判定性的一些定理.
英文摘要：
We prove a result on the representation of squares by second degree polynomials in the field of $p$-adic meromorphic functions in order to solve positively Büchi's $n$ squares problem in this field (that is, the problem of the existence of a constant $M$ such that any sequence $(x_n^2)$ of $M$ - not all constant - squares whose second difference is the constant sequence $(2)$ satisfies $x_n^2=(x+n)^2$ for some $x$). We prove (based on works by Vojta) an analogous result for function fields of characteristic zero, and under a Conjecture by Bombieri, an analogous result for number fields. Using an argument by Büchi, we show how the obtained results improve some theorems about undecidability for the field of $p$-adic meromorphic functions and the ring of $p$-adic entire functions.
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