Weierstrass division set

The Weierstrass Division set is a mathematical theorem from the function theory of several variables. The theorem allows division with remainder with respect to a Weierstraß polynomial .

Introduction and formulation of the sentence

Let it denote the ring of convergent power series around 0. Each can be regarded as an element of by means of the definition . In particular, that is polynomial in included. Therefore one can speak of a polynomial degree . This is especially true for Weierstrass polynomials, that is, polynomials of the form ${\displaystyle {}_{n}{\mathcal {O}}}$ ${\displaystyle f\in {}_{n-1}{\mathcal {O}}}$ ${\displaystyle f(z_{1},\ldots ,z_{n}):=f(z_{1},\ldots ,z_{n-1})}$ ${\displaystyle {}_{n}{\mathcal {O}}}$ ${\displaystyle {}_{n-1}{\mathcal {O}}[z_{n}]}$ ${\displaystyle {}_{n}{\mathcal {O}}}$

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

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

With these terms the following so-called Weierstrasse division theorem applies

• Let it be a Weierstrass polynomial of degree . Then each has a unique representation as ${\displaystyle h\in {}_{n-1}{\mathcal {O}}[z_{n}]}$ ${\displaystyle k\geq 0}$ ${\displaystyle f\in {}_{n}{\mathcal {O}}}$

${\displaystyle f=g\cdot h+r}$   with   , , . ${\displaystyle g\in {}_{n}{\mathcal {O}}}$ ${\displaystyle r\in {}_{n-1}{\mathcal {O}}[z_{n}]}$ ${\displaystyle \mathrm {grad} \,r<k}$

Is , so is . ${\displaystyle f\in {}_{n-1}{\mathcal {O}}[z_{n}]}$ ${\displaystyle g\in {}_{n-1}{\mathcal {O}}[z_{n}]}$

Proof idea

The power series and both converge on a suitable polycircle . Since is a Weierstrass polynomial, one can find such that for all and . The functions are then defined on ${\displaystyle f}$ ${\displaystyle h}$ ${\displaystyle \Delta (0;r_{1},\ldots ,r_{n})}$ ${\displaystyle h}$ ${\displaystyle 0<\delta _{j}<r_{j}}$ ${\displaystyle h(z_{1},\ldots ,z_{n})\not =0}$ ${\displaystyle |z_{1}|<\delta _{1},\ldots |z_{n-1}|<\delta _{n-1}}$ ${\displaystyle |z_{n}|=\delta _{n}}$ ${\displaystyle \Delta (0;\delta _{1},\ldots ,\delta _{n})}$

${\displaystyle g(z_{1},\ldots ,z_{n}):={\frac {1}{2\pi \mathrm {i} }}\int _{|\zeta |=\delta _{n}}{\frac {f(z_{1},\ldots ,z_{n-1},\zeta )}{h(z_{1},\ldots ,z_{n-1},\zeta )}}\cdot {\frac {1}{\zeta -z_{n}}}\mathrm {d} \zeta }$

${\displaystyle r(z_{1},\ldots ,z_{n}):=f(z_{1},\ldots ,z_{n})-g(z_{1},\ldots ,z_{n})\cdot h(z_{1},\ldots ,z_{n})}$,

which can then be shown to provide the unequivocal representation claimed.

The case n = 1

For the Weierstraß polynomial is necessary the normalized monomial and for each one receives the simple relation ${\displaystyle n=1}$ ${\displaystyle h}$ ${\displaystyle z_{1}^{k}}$ ${\displaystyle \textstyle f(z_{1})=\sum _{j=0}^{\infty }a_{j}z_{1}^{j}}$

${\displaystyle f(z_{1})=\sum _{j=k}^{\infty }a_{j}z_{1}^{j}+\sum _{j=0}^{k-1}a_{j}z_{1}^{j}=\sum _{j=0}^{\infty }a_{j+k}z_{1}^{j}\cdot z_{1}^{k}+\sum _{j=0}^{k-1}a_{j}z_{1}^{j}}$.

Therefore the above theorem is only for non-trivial. ${\displaystyle n>1}$

Variant for regular power series

A power series is called regular of order if the holomorphic function has a zero of order . For a Weierstrass polynomials of degree applies , ie Weierstrass polynomials have this Regularitätseigenschaft. Therefore, the following variant of Weierstrasse's division theorem is more general: ${\displaystyle h\in {}_{n}{\mathcal {O}}}$ ${\displaystyle z_{n}}$ ${\displaystyle k}$ ${\displaystyle z_{n}\mapsto h(0,\ldots ,0,z_{n})}$ ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle h(0,\ldots ,0,z_{n})=z_{n}^{k}}$

• It is in regular order . Then each has a unique representation as ${\displaystyle h\in {}_{n-1}{\mathcal {O}}[z_{n}]}$ ${\displaystyle z_{n}}$ ${\displaystyle k\geq 0}$ ${\displaystyle f\in {}_{n}{\mathcal {O}}}$

${\displaystyle f=g\cdot h+r}$   with   , , . ${\displaystyle g\in {}_{n}{\mathcal {O}}}$ ${\displaystyle r\in {}_{n-1}{\mathcal {O}}[z_{n}]}$ ${\displaystyle \mathrm {grad} \,r<k}$

Is , so is . ${\displaystyle f\in {}_{n-1}{\mathcal {O}}[z_{n}]}$ ${\displaystyle g\in {}_{n-1}{\mathcal {O}}[z_{n}]}$

This follows easily from the version given above, because according to Weierstraß's preparatory theorem one can write with a unit and a Weierstraß polynomial . According to the above version of the Division set there are unique , , so that . Then there is a division breakdown of the desired kind. ${\displaystyle h=uh_{0}}$ ${\displaystyle u\in {}_{n}{\mathcal {O}}}$ ${\displaystyle h_{0}\in {}_{n-1}{\mathcal {O}}[z_{n}]}$ ${\displaystyle g\in {}_{n}{\mathcal {O}}}$ ${\displaystyle r\in {}_{n-1}{\mathcal {O}}[z_{n}]}$ ${\displaystyle \mathrm {grad} \,r<k}$ ${\displaystyle f=g\cdot h_{0}+r}$ ${\displaystyle f=g\cdot u^{-1}\cdot u\cdot h_{0}+r=gu\cdot h+r}$

Relationship to the prep kit

From the second version, in which the preparatory sentence was incorporated, the latter can easily be recovered. Is in fact regularly in the order , so there is according to the above sentence , , with . If one evaluates this equation in , it follows ${\displaystyle f\in {}_{n}{\mathcal {O}}}$ ${\displaystyle z_{n}}$ ${\displaystyle k}$ ${\displaystyle g\in {}_{n}{\mathcal {O}}}$ ${\displaystyle r\in {}_{n-1}{\mathcal {O}}[z_{n}]}$ ${\displaystyle \mathrm {grad} \,r<k}$ ${\displaystyle z_{n}^{k}=g\cdot f+r}$ ${\displaystyle (0,\ldots ,0,z_{n})}$

${\displaystyle z_{n}^{k}=g(0,\ldots ,0,z_{n})\cdot f(0,\ldots ,0,z_{n})+\sum _{j=0}^{k-1}r_{j}(0,\ldots ,0)z_{n}^{j}}$.

So all must disappear and must be a unit to maintain the zero order. Therefore is a product of a unit and a Weierstrass polynomial, which ends the derivation of Weierstrass's preparatory theorem from the above version of the division theorem. ${\displaystyle r_{j}(0,\ldots ,0)}$ ${\displaystyle g}$ ${\displaystyle f=g^{-1}\cdot (z_{n}^{k}-r)}$

meaning

Weierstraß's division theorem, together with Weierstraß's preparatory theorem, enables the proof of important properties of the local integrity rings : ${\displaystyle {}_{n}{\mathcal {O}}}$

• ${\displaystyle {}_{n}{\mathcal {O}}}$is a factorial ring .
• ${\displaystyle {}_{n}{\mathcal {O}}}$is a noetherian ring . ( Rückert's basic theorem )
• Every finitely generated module has a free resolution of the length . ( Hilbert's syzygy theorem ) ${\displaystyle {}_{n}{\mathcal {O}}}$ ${\displaystyle \leq n}$

Variant for functions

The previous versions of the division theorem deal with convergent power series around 0, that is, seeds of holomorphic functions around 0. In the following, a variant is presented for functions that are defined in the vicinity of a solid compact polycircle , where stands for the end of the polycircle. re-denote the ring of seeds of holomorphic functions , that is, the set of all holomorphic functions defined in an open neighborhood , two such functions being identified if they match on a common open neighborhood of . Since non-empty interior has, each because of identity set by its very values determined, which means you it has to do with real functions, and defines a norm on . In order to be able to use the same idea of ​​proof as above, the first part of this idea of ​​proof must be included in the requirements of the sentence. This explains the following formulation: ${\displaystyle K={\overline {\Delta }}(0;r_{1},\ldots ,r_{n})\subset \mathbb {C} ^{n}}$ ${\displaystyle {\overline {\Delta }}}$ ${\displaystyle {}_{n}{\mathcal {O}}_{K}}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle f\in {}_{n}{\mathcal {O}}_{K}}$ ${\displaystyle K}$ ${\displaystyle \|f\|_{K}:=\sup\{|f(z)|;z\in K\}}$ ${\displaystyle {}_{n}{\mathcal {O}}_{K}}$

• It is a compact Polykreis, . Is further such that the function of germ in 0 a Weierstrass polynomial of degree respect. , And for each of all the solutions of the condition fulfilled. Then there is a constant such that: ${\displaystyle K={\overline {\Delta }}(0;r_{1},\ldots ,r_{n})\subset \mathbb {C} ^{n}}$ ${\displaystyle K':={\overline {\Delta }}(0;r_{1},\ldots ,r_{n-1})\subset \mathbb {C} ^{n-1}}$ ${\displaystyle h\in {}_{n}{\mathcal {O}}_{K}}$ ${\displaystyle h}$ ${\displaystyle k}$ ${\displaystyle z_{n}}$ ${\displaystyle (a_{1},\ldots ,a_{n-1})\in K'}$ ${\displaystyle k}$ ${\displaystyle h(a_{1},\ldots ,a_{n-1},z_{n})=0}$ ${\displaystyle |z_{n}|<r_{n}}$ ${\displaystyle C>0}$

Each has a unique representation ${\displaystyle f\in {}_{n}{\mathcal {O}}_{K}}$

${\displaystyle f=g\cdot h+r}$

with ,   ${\displaystyle g\in {}_{n}{\mathcal {O}}_{K}}$ ${\displaystyle \|g\|_{K}\leq C\|f\|_{K}}$

and ,   ,   ${\displaystyle r(z_{1},\ldots ,z_{n})=\sum _{j=0}^{k-1}r_{j}(z_{1},\ldots ,z_{n-1})z_{n}^{j}}$ ${\displaystyle r_{j}\in {}_{n-1}{\mathcal {O}}_{K'}}$ ${\displaystyle \|r_{j}\|_{K'}\leq C\|f\|_{K}}$

As mentioned earlier, the proof idea presented above works. Additional work is required to determine the constants that only depend on and . ${\displaystyle K}$ ${\displaystyle h}$ ${\displaystyle C}$

Individual evidence

1. ^ Gunning - Rossi : Analytic functions of several complex variables . Prentice-Hall, 1965, chap. II.B, Theorem 3 (Weierstrass Division Theorem)
2. ↑ Behnke , Thullen : Theory of the functions of several complex variables. Springer-Verlag, 1970, ISBN 3-642-62005-1 , p. 104, appendix to chap. V, §1: The preparatory sentence
3. ^ Gunning-Rossi: Analytic functions of several complex variables . Prentice-Hall, 1965, chap. II.B, Theorem 7.
4. ↑ Jörg Eschmeier: Function theory of several variables. Springer-Verlag, 2017, ISBN 978-3-662-55541-5 , Corollary 4.20.
5. ^ Gunning-Rossi: Analytic functions of several complex variables . Prentice-Hall, 1965, chap. II.B, Theorem 9.
6. ^ Gunning-Rossi: Analytic functions of several complex variables . Prentice-Hall, 1965, chap. II.C, Theorem 2.
7. ^ Gunning-Rossi: Analytic functions of several complex variables . Prentice-Hall, 1965, chap. II.D, Theorem 1. (Extended Weierstrass Division Theorem)