Weierstrass polynomial

The mathematical concept of the Weierstrass polynomial , named after Karl Weierstrass , appears in the function theory of several variables. These are holomorphic functions or function nuclei in a point which, with respect to one of the variables, is a normalized polynomial with coefficients from the ring of holomorphic functions in the other variables, so that the coefficients also disappear at this point.

Definitions

Let it be the ring of convergent power series in . This ring is isomorphic to the ring of the function seeds of holomorphic functions in 0, which is why this concept formation can also be carried out for seeds. Through the injective mapping ${\displaystyle {}_{n}{\mathcal {O}}}$ ${\displaystyle 0\in \mathbb {C} ^{n}}$

${\displaystyle {}_{n-1}{\mathcal {O}}\rightarrow {}_{n}{\mathcal {O}},\quad f\mapsto {\tilde {f}},\quad {\tilde {f}}(z_{1},\ldots ,z_{n}):=f(z_{1},\ldots ,z_{n-1})}$

is understood as a sub-ring of . This means that it becomes an element because the last variable is simply ignored when evaluating the variable . ${\displaystyle {}_{n-1}{\mathcal {O}}}$ ${\displaystyle {}_{n}{\mathcal {O}}}$ ${\displaystyle f\in {}_{n-1}{\mathcal {O}}}$ ${\displaystyle {}_{n}{\mathcal {O}}}$ ${\displaystyle (z_{1},\ldots ,z_{n})}$

The variable is itself a polynomial and therefore an element of . Adjoint to the sub-ring , one obtains the polynomial ring with coefficients , and one has the inclusions ${\displaystyle z_{n}}$ ${\displaystyle {}_{n}{\mathcal {O}}\setminus {}_{n-1}{\mathcal {O}}}$ ${\displaystyle z_{n}}$ ${\displaystyle {}_{n-1}{\mathcal {O}}}$ ${\displaystyle {}_{n-1}{\mathcal {O}}[z_{n}]}$ ${\displaystyle {}_{n-1}{\mathcal {O}}}$

${\displaystyle {}_{n-1}{\mathcal {O}}\subset {}_{n-1}{\mathcal {O}}[z_{n}]\subset {}_{n}{\mathcal {O}}}$.

Each element from has a unique representation ${\displaystyle f\in {}_{n-1}{\mathcal {O}}[z_{n}]}$

${\displaystyle f(z_{1},\ldots ,z_{n})=a_{m}(z_{1},\ldots ,z_{n-1})\cdot z_{n}^{m}+a_{m-1}(z_{1},\ldots ,z_{n-1})\cdot z_{n}^{m-1}+\ldots +a_{1}(z_{1},\ldots ,z_{n-1})\cdot z_{n}+a_{0}(z_{1},\ldots ,z_{n-1})}$

with convergent power series . ${\displaystyle a_{0},a_{1},\ldots ,a_{m-1},a_{m}\in {}_{n-1}{\mathcal {O}}}$

Such an element is called a Weierstrass polynomial, if

• ${\displaystyle a_{m}}$is the constant one function, that is, is a normalized polynomial over , and ${\displaystyle f}$ ${\displaystyle {}_{n-1}{\mathcal {O}}}$
• ${\displaystyle a_{k}(0,\ldots ,0)=0}$for everyone . ${\displaystyle k=0,\ldots ,m-1}$

Examples

• For functions of a variable, a Weierstraß polynomial is nothing more than a normalized monomial , that is, of the form . ${\displaystyle z_{1}^{m}}$

• The polynomial is not a Weierstraß polynomial because it is not normalized. ${\displaystyle z_{1}z_{2}\in {}_{1}{\mathcal {O}}[z_{2}]}$

• The polynomial is also not a Weierstraß polynomial, since the coefficient of does not vanish into 0. ${\displaystyle z_{2}^{3}+\exp(z_{1})z_{2}+z_{1}^{4}\in {}_{1}{\mathcal {O}}[z_{2}]}$ ${\displaystyle z_{2}}$

• The polynomial is a Weierstraß polynomial. ${\displaystyle z_{2}^{3}+\sin(z_{1})z_{2}+z_{1}^{4}\in {}_{1}{\mathcal {O}}[z_{2}]}$

comment

Weierstraß polynomials play an important role in the function theory of several variables because they allow a kind of division with remainder , as occurs in Weierstraß's division theorem . The irreducible elements in the ring are essentially the Weierstraß polynomials which are irreducible in the polynomial ring . ${\displaystyle {}_{n}{\mathcal {O}}}$ ${\displaystyle {}_{n-1}{\mathcal {O}}[z_{n}]}$

See also

• Weierstrasse preparatory sentence

Individual evidence

1. ^ Gunning - Rossi : Analytic functions of several complex variables . Prentice-Hall, 1965, chap. II .B, definition 1.
2. ↑ Jörg Eschmeier: Function theory of several variables. Springer-Verlag, 2017, ISBN 978-3-662-55541-5 , definition 4.18.
3. ^ Gunning-Rossi: Analytic functions of several complex variables . Prentice-Hall, 1965, chap. II.B, Theorem 7.