Several complex variables and analytic spaces Definition

Weierstrass polynomial

Builds on Ring of convergent power series Monic polynomial

Definition

Let $n \ge 1$ , write $z' = (z_1, \dots, z_{n-1})$ , and let $\mathcal{O}_{n-1} = \C\{z'\}$ be the ring of convergent power series in $n - 1$ variables. Let $d \ge 0$ be an integer. A Weierstrass polynomial of degree $d$ in $z_n$ is a monic polynomial $P(z', z_n) = z_n^d + a_{d-1}(z')\, z_n^{d-1} + \cdots + a_1(z')\, z_n + a_0(z')$ with coefficients $a_0, \dots, a_{d-1} \in \mathcal{O}_{n-1}$ such that $a_j(0) = 0$ for every $j$ .

Remarks

Equivalently, $P \in \mathcal{O}_{n-1}[z_n]$ is monic of degree $d$ and $P(0, z_n) = z_n^d$ . The only Weierstrass polynomial of degree $0$ is $P = 1$ . Example: $P = z_2^3 - z_1 z_2 + z_1^2$ is a Weierstrass polynomial of degree $3$ in $z_2$ , since $a_2 = 0$ , $a_1 = -z_1$ and $a_0 = z_1^2$ all vanish at $0$ . Non-example: $Q = z_2^2 + z_2 + z_1$ is monic in $z_2$ , but its coefficient $a_1 = 1$ does not vanish at $0$ ; indeed $Q(0, z_2) = z_2^2 + z_2 \ne z_2^2$ . The same definition is used with $\mathcal{O}_{n-1}$ replaced by the ring of holomorphic functions on a neighbourhood of $0 \in \C^{n-1}$ .

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Tag	`RD7DVC2NJ6`
URI	`m2://RD7DVC2NJ6`
Kind	definition
Writing	W0 stub
MSC	32B05, 13J07
Defines	Weierstrass polynomial
Uses	Ring of convergent power series, Monic polynomial

Weierstrass polynomial

Definition

Remarks

Reading path

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