padic radius of convergence of power series solutions of padic differential equations.
Project/Area Number  09640079 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Maizuru National College of Technology 
Principal Investigator 
SETOYANAGI Minoru Maizuru National College of Technology, Department of Natural Science, Associate Professor, 自然科学科, 助教授 (20196976)

Project Period (FY) 
1997 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1999 : ¥200,000 (Direct Cost : ¥200,000)
Fiscal Year 1998 : ¥200,000 (Direct Cost : ¥200,000)
Fiscal Year 1997 : ¥300,000 (Direct Cost : ¥300,000)

Keywords  padic differential equation / power series solution / padic radius of convergence / padic Liouville number / P進微分方程式 / 特異点 / P進収束半径 / 巾級教解 / 巾級数解 
Research Abstract 
1. We determine the greatest lower bound of the radii of convergence of power series solutions at an ordinary point in general, I.e., every solution of a padic linear differential equation whose coefficients are convergent power series has the radius R of convergence satisfying the inequality R【greater than or equal】kィイD2pィエD2pィイD11/(p1)ィエD1, where kィイD2pィエD2 is determined by the equation. Moreover, there is an equation and its solution such that the equality R = kィイD2pィエD2pィイD11/(p1)ィエD1 holds. 2. We determine the radius RィイD2αィエD2 of convergence of the binomial series (1+x)ィイD1αィエD1. (1) RィイD2αィエD2 is infinity if α is non negative integer. (2) RィイD2αィエD2 = 1 if α is padic integer but non negative integer. (3) RィイD2αィエD2 = pィイD1w(α)(1/(p1))ィエD1 if α is not a padic integer and 1 【greater than or equal】α. (4) RィイD2αィエD2 = (1/α)pィイD11/(p1)ィエD1 if α>1. 3. Let x=0 be a singular point of the homogenous linear differential equation. If each root of its inditial equation at the singular point is padically nonLiouville (in the sense of Schikhof), then Laurent series solutions at x=0 converge in the punctured disk 0 <xィイD2pィエD2 <μィイD2pィエD2 A(f) pィイD1w(f)ィエD1, where μィイD2pィエD2 is determined by the differential equation ; A(f) and w(f) are numbers determined by the polynomial f(x)=0. Moreover, there is an example such that the radius of convergence is equal to μィイD2pィエD2 A(f)pィイD1w(f)ィエD1.

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Research Products
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