Prime and almost prime integral points on principal homogeneous spaces

Details Prime and almost prime integral points on principal homogeneous spaces Prime and almost prime integral points on principal homogeneous spaces

2009-02-04

arxiv.org/abs/0902.0692v1

Mathematics

Number Theory

Scientific paper

We develop the affine sieve in the context of orbits of congruence subgroups of semi-simple groups acting linearly on affine space. In particular we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors. In many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable and the orbit is the integers. When the orbit is the set of integral matrices of a fixed determinant we obtain a sharp result for the saturation number, and thus establish the Zariski density of matrices all of whose entries are prime numbers. Among the key tools used are explicit approximations to the generalized Ramanujan conjectures for such groups and sharp and uniform counting of points on such orbits when ordered by various norms.

Affiliated with

Also associated with

No associations

Rating

Prime and almost prime integral points on principal homogeneous spaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Prime and almost prime integral points on principal homogeneous spaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Prime and almost prime integral points on principal homogeneous spaces will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-553361