scholarly journals Restrictions of Free Arrangements and the Division Theorem

2017 ◽  
pp. 389-401 ◽  
Author(s):  
Takuro Abe
Keyword(s):  
Division Theorem

Weierstrass Division in Quasianalytic Local Rings

1976 ◽  
Vol 28 (5) ◽  
pp. 938-953 ◽  
Author(s):  
C. L. Childress
Keyword(s):  
Local Ring ◽  
Analytic Functions ◽  
Local Rings ◽  
Division Theorem ◽  
Background Material

In this paper we consider the problem of extending the Weierstrass division theorem to quasianalytic local rings of germs of functions of k real variables which properly contain the local ring of germs of analytic functions. After some background material (§ 2) and some technical preliminaries (§ 3), we show by examples that the so-called generic division theorem fails in such rings if k ≧ 1 and that the Weierstrass division theorem fails in such rings if k ≧ 2 (§ 4).

Applications of the division theorem inH s

1996 ◽  
Vol 8 (3) ◽  
pp. 407-425 ◽  
Author(s):  
Satyanad Kichenassamy
Keyword(s):  
Division Theorem

scholarly journals The Malgrange-Mather division theorem

Topology ◽  
1977 ◽  
Vol 16 (4) ◽  
pp. 395-401 ◽  
Author(s):  
Pierre Milman
Keyword(s):  
Division Theorem

A Precise L 2 Division Theorem

2002 ◽  
pp. 185-191 ◽  
Author(s):  
Takeo Ohsawa
Keyword(s):  
Division Theorem

scholarly journals Local zero estimates and effective division in rings of algebraic power series

2018 ◽  
Vol 2018 (737) ◽  
pp. 111-160 ◽  
Author(s):  
Guillaume Rond
Keyword(s):  
Power Series ◽  
Necessary Condition ◽  
Effective Solution ◽  
Gap Theorem ◽  
Division Theorem ◽  
Ideal Membership ◽  
Formal Power ◽  
Algebraic Power Series ◽  
Ideal Membership Problem ◽  
The Ideal

AbstractWe give a necessary condition for algebraicity of finite modules over the ring of formal power series. This condition is given in terms of local zero estimates. In fact, we show that this condition is also sufficient when the module is a ring with some additional properties. To prove this result we show an effective Weierstrass Division Theorem and an effective solution to the Ideal Membership Problem in rings of algebraic power series. Finally, we apply these results to prove a gap theorem for power series which are remainders of the Grauert–Hironaka–Galligo Division Theorem.

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