Forms representing forms and linear spaces on hypersurfaces

Abstract

For a given set of forms ψ(1), …, ψ(R)∈ℤ[t1, …, tm] of degree d we prove a Hasse principle for representations of the shape F ( ρ ) ( t 1 x 1 + ⋯ + t m x m ) = ψ ( ρ ) ( t 1 , … , t m ) , 1 ⩽ ρ ⩽ R by general forms F(1), …, F(R)∈ℤ[x1, …, xs] of the same degree, provided that s≫R2md and the forms F(ρ) are ‘sufficiently non-singular’. This result is then used to derive asymptotical behaviour of the number of m-dimensional linear spaces contained in the intersection of the F(ρ) if the degree is odd. A further application dispenses with the non-singularity condition and establishes the existence of m-dimensional linear spaces on the intersection of R cubic forms if the number s of variables asymptotically exceeds R6+m3 R3. Finally, we briefly consider linear spaces on small systems of quintic equations.