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A new \(p\)-adic method for proving irrationality and transcendence results. (English) Zbl 0679.10025

This surprising note links a fairly technical \(p\)-adic criterion for rationality due to the authors [‘Rational solutions of linear differential equations’, J. Aust. Math. Soc., Ser. A 46, No. 2, 184–196 (1989; Zbl 0695.12019)] for which the special case required here is, however, fairly straightforward and is detailed independently, and the observation that the Lindemann-Weierstraß Theorem is entailed by the following result: Suppose \(u\) is a formal power series over \({\mathbb Q}\) and has nonzero radius of convergence. Then the rationality of \(w(x)=x^ 2u'(x)+(x-1)u(x)\) entails the rationality of \(u\). The apparently simple principle is that the Lindemann-Weierstraß Theorem asserts exactly that a finite sum \(\sum\) \(A_ i \exp \alpha_ ix\) with algebraic parameters does not vanish at \(x=1\) (except of course in trivial cases) and that such a sum is a divisor (in the ring of entire functions) of a Taylor series with rational coefficients.
The fact that \(u\) be defined over \({\mathbb Q}\) is essential to the argument as detailed, though, of course, it suffices for the field of definition to be some number field. Subsequent to presentation of the argument it was pointed out by Beukers that its \(p\)-adic features are not of the essence. Nevertheless, it remains of significant interest to understand just what conditions are required on \(u\) to permit the argument to succeed and just what rôle is played by the \(p\)-adic features.

MSC:

11J81 Transcendence (general theory)
11J72 Irrationality; linear independence over a field
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
12H20 Abstract differential equations

Citations:

Zbl 0695.12019
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