Forms representing forms: the definite case

Let ψ and F be positive‐definite forms with integral coefficients of equal degree. Using the circle method, we establish an asymptotic formula for the number of identical representations of ψ by F , provided that ψ is everywhere locally representable and the number of variables of F is large enough. In the quadratic case, this supersedes a recent result due to Dietmann and Harvey. Another application addresses the number of primitive linear spaces contained in a hypersurface.