A note on p -adic solubility for forms in many variables

Abstract

Abstract. By adopting a new approach to the analysis of the density of p-adic solutions arising in applications of the circle method, we show that under modest conditions the existence of non-trivial p-adic solutions suffices to establish positivity of the singular series. This improves on earlier approaches due to Davenport, Schmidt and others, and allows us to establish an asymptotic formula for the number of simultaneous zeros of non-singular pairs of cubic forms in at least 131 variables. As a by-product, we obtain a version of Hensel's Lemma for linear spaces.

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

“…though we note that for generic forms F this bound may be improved somewhat by earlier work of the author (see [3,Corollary 1]). …”

Simultaneous Additive Equations: Repeated and Differing Degrees

“…dδ (3) , and after t iterations we obtain convergence if u h > 1 2 k̟ h (1 h t). Furthermore, it is clear that under the same condition one has…”

“…Unfortunately, for other situations we do not obtain equally strong results, largely due to the lack of sufficiently powerful mean values. We also note that it may be possible to remove the explicit assumption of non-singularity for the real and p-adic solutions by adapting work of the first author [3]. We intend to pursue some of these refinements in future papers.…”

Rational lines on cubic hypersurfaces

“…Observe that, unlike in Wooley's work, we require a smoothness condition, although this may be somewhat relaxed at the cost of inflating the number of variables correspondingly (see e.g. the argument in Section 4 of [2]). Just like in the earlier work, we approach the problem by first fixing one rational point on the hypersurface, and then constructing a second point with the property that the line defined by these two points is contained in the hypersurface.…”