A Gap Theorem for Differentially Algebraic Power Series

Number Theory ◽  
1991 ◽  
pp. 211-214
Author(s):  
Leonard Lipshitz ◽  
Lee A. Rubel
Keyword(s):  
Power Series ◽  
Gap Theorem ◽  
Algebraic Power Series

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.

On n-algebraic power series having polynomial growths11

2000 ◽  
Vol 28 (7) ◽  
pp. 3253-3264
Author(s):  
Juha Honkala
Keyword(s):  
Power Series ◽  
Algebraic Power Series

scholarly journals REMARKS ON EISENSTEIN

2013 ◽  
Vol 94 (2) ◽  
pp. 158-180 ◽  
Author(s):  
YURI BILU ◽  
ALEXANDER BORICHEV
Keyword(s):  
Power Series ◽  
Number Field ◽  
Laurent Series ◽  
Quantitative Version ◽  
Algebraic Power Series

AbstractWe obtain a fully explicit quantitative version of the Eisenstein theorem on algebraic power series which is more suitable for certain applications than the existing version due to Dwork, Robba, Schmidt and van der Poorten. We also treat ramified series and Laurent series, and we demonstrate some applications; for instance, we estimate the discriminant of the number field generated by the coefficients.

A Gap Theorem for Power Series

1938 ◽  
Vol s1-13 (4) ◽  
pp. 295-301
Author(s):  
J. M. Whittaker
Keyword(s):  
Power Series ◽  
Gap Theorem

Continued Fractions of Algebraic Power Series in Characteristic 2

1976 ◽  
Vol 103 (3) ◽  
pp. 593 ◽  
Author(s):  
Leonard E. Baum ◽  
Melvin M. Sweet
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Characteristic 2 ◽  
Algebraic Power Series
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