Linear and algebraic independence of logarithms. (Indépendance linéaire et algébrique de logarithmes.) (French) Zbl 0629.10028

Hellegouarch, Yves

Groupe Étude Théor. Anal. Nombres 1re-2e années 1984/1985, Exp. No. 20, 8 p. (1985).

Let \(k\) be a field of characteristic 0 and let \(P_1(X),\ldots, P_r(X)\) be nonconstant polynomials two by two prime with each other, such that \(P_i(0)=1\) whenever \(i=1,\ldots,r\). The author here proves by elementary processes the formal series \(\text{Log}(P_i(X)=-\sum^{\infty}_{n=1}(1- P(X))n/n\) \((1\leq i\leq r)\) are first linearly independent over \(k(X)\) and second, are algebraically independent over \(k(X)\). The result was already known (the author cites J. Ax and A. Ostrowski). Here he gives a new method that only involves elementary processes, particularly the comma he had previously defined for a multiplicative group \(G\) in \(\mathbb{Q}\), and he generalizes it by taking an absolute value on \(k((X))\) trivial on \(k\).

Reviewer: Alain Escassut

MSC:

11J72	Irrationality; linear independence over a field
11J86	Linear forms in logarithms; Baker’s method
13F25	Formal power series rings

Keywords:

linear independence; algebraic independence; formal series; comma