Theorem of preparation of Weierstrass

That is to say $\backslash \; mathbb\; \{K\}$ a complete body valué not archimédien of null characteristic, associated with a valuation W and an absolute value $|.\; |$. That is to say $c\; >\; 0$ and $P\; (X)\; =\; \backslash \; sum\_\; \{k=0\}\; ^\; \{m\}\; a\_k\; X^k\; \backslash \; in\; \backslash \; mathbb\; \{K\}$. One notes $\backslash \; lVert\; P\; (X)\; \backslash \; rVert\_c\; =\; \backslash \; max\_\; \{0\; \backslash \; Leq\; I\; \backslash \; Leq\; m\}\; (|a\_i|c^i)$.

If there exists an entirety $N\; \backslash \; in\; \backslash \; \{1,\dots ,\; M-1\; \backslash \}$ for which $\backslash \; lVert\; P\; (X)\; \backslash \; rVert\_c\; =\; |a\_N|c^N$ and $\backslash \; lVert\; P\; (X)\; \backslash \; rVert\_c\; >\; |a\_i|c^i$ for all $i\; >\; N$, then:

(1) There exist two polynomials $Q\; (X),\; R\; (X)\; \backslash \; in\; \backslash \; mathbb\; \{K\}$ such as $\backslash \; deg\; (Q)\; =\; N$, $\backslash \; deg\; (R)\; =\; m\; -\; N$ and $P\; (X)\; =\; Q\; (X)\; R\; (X)$.

(2) Moreover, there is $\backslash \; lVert\; Q\; (X)\; \backslash \; rVert\_c\; =\; \backslash \; lVert\; P\; (X)\; \backslash \; rVert\_c$ and .

Category: Algebraic theory of the numbers Weierstrass

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