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 $ψ$ 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.

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