Abstract. For a given set of forms ψ (1) , . . . , ψ (R) ∈ Z[t 1 , . . . , tm] of degree d we prove a Hasse principle for representations of the shape. . , xs] of the same degree, provided that s ≫ R 2 m d 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 R 6 + m 3 R 3 . Finally, we briefly consider linear spaces on small systems of quintic equations.

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.

Abstract. We employ a generalised version of Heath-Brown's square sieve in order to establish an asymptotic estimate of the number of solutions a, b to the equations a + b = n and a − b = n, where a is k-free and b is l-free. This is the first time that this problem has been studied with distinct powers k and l.

Abstract. We obtain bounds for the number of variables required to establish Hasse principles, both for existence of solutions and for asymptotic formulae, for systems of additive equations containing forms of di ering degree but also multiple forms of like degree. Apart from the very general estimates of Schmidt and Browning-HeathBrown, which give weak results when specialized to the diagonal situation, this is the rst result on such "hybrid" systems. We also obtain specialised results for systems of quadratic and cubic forms, where we are able to take advantage of some of the stronger methods available in that setting. In particular, we achieve essentially square root cancellation for systems consisting of one cubic and r quadratic equations.

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