p-adic radius of convergence of power series solutions of p-adic differential equations.
Project/Area Number |
09640079
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Research Category |
Grant-in-Aid for Scientific Research (C)
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Allocation Type | Single-year Grants |
Section | 一般 |
Research Field |
Algebra
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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)
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Project Period (FY) |
1997 – 1999
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Project Status |
Completed (Fiscal Year 1999)
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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)
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Keywords | p-adic differential equation / power series solution / p-adic radius of convergence / p-adic 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 p-adic 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D1-1/(p-1)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D1-1/(p-1)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 p-adic integer but non negative integer. (3) RィイD2αィエD2 = pィイD1w(α)-(1/(p-1))ィエD1 if α is not a p-adic integer and 1 【greater than or equal】|α|. (4) RィイD2αィエD2 = (1/|α|)pィイD1-1/(p-1)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 p-adically non-Liouville (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D1-w(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D1-w(f)ィエD1.
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Report
(4 results)
Research Products
(3 results)