Forms representing forms: the definite case

Abstract

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.

The density of rational lines on hypersurfaces: a bihomogeneous perspective

“…Theorem 1.4 by a product of local densities as in (7.14). This establishes Theorem 1.1 for all ψ ≤ ψ 0 , while for ψ 0 ≤ ψ ≤ 1 the corresponding result follows from Theorem 2.1 in [9]. Finally, we recall that we need ρ ≥ 1 2 d(d + 1) + 1 and note that…”

“…When F is a cubic form, recent work of the author jointly with Dietmann [12] shows that (1.1) has non-trivial rational solutions whenever n ≥ 29, but that there may not be any rational solutions when n = 11 or lower. For more general settings, (1.1) has been investigated in a series of papers by the present author [8][9][10][11]. We note at this point that, in order to strictly count lines, we would have to exclude those solutions of (1.1) where x and y are proportional.…”

“…The main new input in our present work is therefore our treatment of the situation when Y is vastly smaller than X . Unlike in our former work in [8,9], where we allowed the variables x, y to vary independently, we pursue a slicing approach inspired by [23] in which we fix a point y ∈ U(Z) and then investigate the number N y (X ; U) of points x ∈ U(Z) ∩ [−X , X ] n for which (1.1) is satisfied with that particular value y. We then have…”

“…The reader may wonder how the lower bound on n compares with that which can be extracted from [9,Theorem 2.1]. In that result, the bound on the number of variables in the case when ψ is small can be written in terms of ψ as…”

On the number of linear spaces on hypersurfaces with a prescribed discriminant

“…. , x s ] be a form of degree d. In previous work [2,4] we investigated the number of rational linear spaces of dimension m contained in the hypersurface given by . for some non-negative constants χ ∞ and χ p characterising the density of solutions over the local fields R and Q p , respectively, provided that…”