A Note on p-adic Linear Differential Equations

✍ Scribed by Abdelbaki Boutabaa

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 98 KB
• Volume: 87
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.2000.2602

✦ Synopsis

Dekker, New York), we showed that if a function f (x) meromorphic in all C p is a solution of a homogeneous linear differential equation (E) with coefficients in Q (x), then f # C p (x). Here we show that this conclusion is false in the case where (E) is with coefficients not in Q (x).

2001 Academic Press

Theorem A. Suppose that Eq. (E s ) is such that Q 0 (x), ..., Q s (x) # Q (x) and let y(x) # M(C p ) be a solution of (E s ). Then y(x) # C p (x).

Here we will show that, if Q 0 (x), ..., Q s (x) are not all in Q (x), the conclusion in Theorem A may be false (see Theorem B). This answers a question posed by P. C. Hu and C. C. Yang [9].