超度量根计数
Ultrametric Root Counting
摘要：
设$K$是具有离散赋值的完备非阿基米德域，$f\in K[X]$是具有非零判别式的多项式，$A$是$K$的赋值环，$\M$是$A$的极大理想.本文的第一个主要结果是一个Hensel引理，连接$f$的根数与其减少模$\M$的幂的根数。然后，我们定义了一个条件- { \em正则性} -产生一个简单的方法来计算$K$中$f$的确切根数。特别是，我们表明，规律性意味着根的数量的$f$等于某些二项式的根的数量的总和来自牛顿多边形。
英文摘要：
Let $K$ be a complete non-archimedean field with a discrete valuation, $f\in K[X]$ a polynomial with non-vanishing discriminant, $A$ the valuation ring of $K$, and $\M$ the maximal ideal of $A$. The first main result of this paper is a reformulation of Hensel's lemma that connects the number of roots of $f$ with the number of roots of its reduction modulo a power of $\M$. We then define a condition --- {\em regularity} --- that yields a simple method to compute the exact number of roots of $f$ in $K$. In particular, we show that regularity implies that the number of roots of $f$ equals the sum of the numbers of roots of certain binomials derived from the Newton polygon.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
