Multiplicative relations of points on algebraic groups

Our aim here is to restructure the area of multiplicative relations on points and congruences, by proposing a novel conjecture in the context of general reductive linear algebraic groups. To support our conjecture we check it in a few elementary but new cases, and claim this extends classical work in number theory on multiplicative relations on points and congruences, initiated by Skolem and Schinzel, which we rephrase group-theoretically as Hasse principles on commutative linear algebraic groups, or tori, so that a part of it becomes the abelian case of our conjecture. Our conjecture can then be viewed as an extension to general—not necessarily commutative—reductive linear algebraic groups of a part of Schinzel’s result. We relate it to the Erdős support problem. To motivate our conjecture from another perspective we note that analogues have been extensively developed for abelian varieties. We give a short account of this, and state a question on the “detecting linear dependence” problem.