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Let G be the product of an abelian variety and a torus defined over a numberfield K. The aim of this paper is detecting the dependence among some givenrational points of G by studying their reductions modulo all primes of K. Weshow that if some simple conditions on the order of the reductions of thepoints are satisfied then there must be a dependency relation over the ring ofK-endomorphisms of G. We generalize Larsen's result on the support problem toseveral points on products of abelian varieties and tori.

THE MULTILINEAR SUPPORT PROBLEM FOR PRODUCTS OF ABELIAN VARIETIES AND TORI