Abstract
We first fix some notation:
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1.
k: field, complete with respect to a non-trivial absolute value. k[[X1,...,X n ]]: formal power series in n variables X1,...,X n .
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2.
We use:
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a.
Greek letters α, β for n-tuples as α = (α1,...,αn), αi ≥ 0, ∈ Z.
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b.
Latin letters r, s for n-tuples as r = (r1,...,r n ), r i > 0, ∈ R.
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c.
Latin letters x, y for n-tuples as x = (x1,..., x n ), x i ∈ k.
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a.
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3.
We set:
$$ \begin{array}{*{20}{c}} {{{r}^{\alpha }}=r_{1}^{{{{\alpha }_{1}}}}\cdots r_{n}^{{{{\alpha }_{n}}}}} \\ {{{x}^{\alpha }}=x_{1}^{{{{\alpha }_{1}}}}\cdots x_{n}^{{{{\alpha }_{n}}}}} \\ {{{X}^{\alpha }}=X_{1}^{{{{\alpha }_{1}}}}\cdots X_{n}^{{{{\alpha }_{n}}}}} \\ {\left| \alpha \right|=\sum {{{\alpha }_{i}}} } \\ {\alpha !=\prod {{{\alpha }_{i}}!} } \\ {\left( {\begin{array}{*{20}{c}} \alpha \\ \beta \\ \end{array}} \right)=\frac{{\alpha !}}{{\beta !\left( {\alpha -\beta } \right)!}}} \\ \end{array} $$ -
4.
We define: |x| ≤ r (resp. |x| < r) ⇔ |x i | ≤ r i (resp. |x i | < r i ), 1 ≤ i ≤ n. We define similarly r′ ≤ r, r′ < r, α′ ≤ α, and α′ < α.
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5.
We set:
P(r)(x) = { y : |y - x| ≤ r } = Polydisk of radius r about x
P0(r)(x) = { y : |y - x| < r } = Strict polydisk of radiues r about x.
P(r) = P(r)(0)
P0(r) = P0(r)(0).
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© 1992 Springer-Verlag Berlin Heidelberg
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Serre, JP. (1992). Analytic Functions. In: Lie Algebras and Lie Groups. Lecture Notes in Mathematics, vol 1500. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70634-2_9
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DOI: https://doi.org/10.1007/978-3-540-70634-2_9
Publisher Name: Springer, Berlin, Heidelberg
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Online ISBN: 978-3-540-70634-2
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