Linear spaces on hypersurfaces over number fields

TL;DR

This paper proves an analytic Hasse principle for linear spaces on hypersurfaces over number fields, extending known results over Q, and shows conditions under which such hypersurfaces are K-unirational.

Contribution

It establishes an analytic Hasse principle for linear spaces on hypersurfaces over algebraic extensions of Q, matching variable requirements known over Q, and applies this to K-unirationality of hypersurfaces.

Findings

Proved an analytic Hasse principle for linear spaces over number fields.

Demonstrated K-unirationality for certain smooth hypersurfaces.

Extended known variable bounds from Q to algebraic extensions.

Abstract

We establish an analytic Hasse principle for linear spaces of affine dimension m on a complete intersection over an algebraic field extension K of Q. The number of variables required to do this is no larger than what is known for the analogous problem over Q. As an application we show that any smooth hypersurface over K whose dimension is large enough in terms of the degree is K-unirational, provided that either the degree is odd or K is totally imaginary.