The density of rational lines on hypersurfaces: A bihomogeneous perspective

Abstract: Let

F

be a non-singular homogeneous polynomial of degree

d

in

n

variables. We give an asymptotic formula of the pairs of integer points

(m a t h b f x, m a t h b f y)

with

| m a t h b f x | l e X

and

| m a t h b f y | l e Y

which generate a line lying in the hypersurface defined by

F

, provided that

n > 2^{d - 1} d^{4} (d + 1) (d + 2)

. In particular, by restricting to Zariski-open subsets we are able to avoid imposing any conditions on the relative sizes of

X

and

Y

.

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