Quadratic relations between logarithms of algebraic numbers
Damien Roy, Michel Waldschmidt · Proceedings of the Japan Academy Series A Mathematical Sciences · 1995
So far, the four exponentials conjecture has P Q[X1,..., Xn] of degree <--2 and let been solved only in one special case, namely be a point in V with coordinates in Assume that when the transcendence degree of the field which the field Q(/21, /2,) has transcendence degree 1 is spanned by the four logarithms is 1. We pro- over Q. Then (/21 /2,,) is contained in a vector duce a new proof of this statement, and we subspace of C n which is defined over Q and conannounce a generalization: we replace the deter- rained in V. minant of a 2 2 matrix by any homogeneous Theorem 1 is the special case of Theorem 2 polynomial of degree 2. when P is X1X4 X2Xa with n 4. 1. The results. The following statement 2. Wirsing’s theorem. When ce is a cornprovides a solution of the four exponentials con- plex algebraic number of degree d [Q(c) Q], jecture in transcendence degree 1. we denote by M(ce) its Mahler’s measure, which Theorem 1. Let x and x2 be two complex is related to its absolute logarithmic height numbers which are linearly independent over Q, and h(c0 by similarly let y, yz be two-linearly independent dh (ce) log M