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Isolated points on fibers of affinoid varieties. (English) Zbl 0636.14006

Let K be a field, complete with respect to a non-trivial, non-Archimedean valuation such that [K:K \(p]<\infty\) if \(char(K)=p>0\). Let V be the valuation ring of K and let \(y=(y_ 1,...,y_ m)\), \(x=(x_ 1,...,x_ n)\) and \(K<y,x>=\{f=\sum_{i,j}a_{ij}y\quad ix\quad j\in K[[ y,x]]| \quad | a_{ij}| \to O\quad as\quad | i| +| j| \to \infty \}\) be the ring of strictly convergent power series over K. Let \(f_ 1,..,f_ N\in K<y,x>\). For every \(a\in V\) m define the fiber \(F_ a=\{b\in V\quad n| \quad f_ i(a,b)=O\quad for\quad i=1,...,N\}.\) The main result of the paper shows that there is a bound c such that for all \(a\in V\) m, \(F_ a\) contains at most c isolated points. In particular, if \(F_ a\) is finite, \(F_ a\) contains at most c points. This result is then extended for subanalytic sets.
Reviewer: L.Bădescu

MSC:

14G20 Local ground fields in algebraic geometry
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14A05 Relevant commutative algebra
14B05 Singularities in algebraic geometry
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