Forms Represented by Linear Forms

This paper is the third in a series in which the author investigates the question of representation of forms by linear forms. Whereas in the first two treatments the proportion of forms F of degree 3 (resp. degree d) which can be written as a sum of two cubes (resp. d-th powers) of linear forms with algebraic coefficients is determined, the generalization now consists in allowing more general expressions of degree d in two linear forms. The main result is thus to give an asymptotic formula, in terms of their height, for the number or decomposable forms that have a representation where f is some fixed homogeneous polynomial and L ₁, L ₂ are linear forms. This is achieved by analyzing some p-adic and archimedean absolute value inequalities combined methods of the geometry of numbers.Received May 24, 2000; in final form January 20, 2003Published online October 24, 2003