Lower bounds of heights of points on hypersurfaces.

In ``Positive line bundles on arithmetic varieties" [J. Am. Math. Soc. 8, 187-221 (1995; Zbl 0861.14018)], \it S. Zhang proved that if X is a subvariety of a linear torus defined over a number field which doesn't contain a translate of a subtorus by a torsion point, then there exists a positive constant c for which X has only finitely many algebraic points of height less than c. On taking c sufficiently small, one can assume these finitely many points to have height zero. Since then, a number of authors [\it E. Bombieri and \it U. Zannier, Int. Math. Res. Not. 1995, No. 7, 333-347 (1995; Zbl 0848.11030), \it W. M. Schmidt, Proc. Am. Math. Soc. 124, 3003-3013 (1996; Zbl 0867.11046) and the paper under review)] have given more elementary proofs of special cases of Zhang's result which also give explicit values for the constant c, and the paper under review gives the best such values. The authors also mention a number of applications of these results