Re: ranges of integer polynomials

On Sun, 07 Oct 2007 09:52:54 -0700, adler.math@xxxxxxxxx wrote:

On Oct 7, 9:16 am, quasi <qu...@xxxxxxxx> wrote:

What did I write? You could have quoted at least something.

Well, at least you included the username of who you were replying to.

In dealing with density questions, we may or may not care whether the
set is recursively recoverable from the range.

Right, since we are only asking about density.

Let N be an abbreviation for N_1^2 + N_2^2 +N_3^2 + N_4^2, and let P
be a Matijasevic polynomial for some set E of positive integers.

Let Q = (N+1) (1 + (N + 3)P^2). Then the range of Q is E union T,
where T is a set of density 0. We can arrange for E to have any
(constructible) desired density d, which will then also be the density
of the range of Q.

Right -- nice.

It looks as if for "general" questions, positive polynomials are not
much better behaved than polynomials. However, the many questions of
number-theoretic character that you raised still remain interesting,
and mostly probably difficult.

Most of them were not difficult. Half of them, 6 out of 12, are
already solved. In fact, given that you just solved one of the density
questions, that makes 7 out of 12. The other density question will
probably be quickly resolved using a similar argument, which would
then make 8, thus leaving only 4 problems unresolved of the original
12. Those last 4 might be a little more difficult, but the truly
difficult ones are not yet posed. They are waiting in the wings.

With Diophantine sets, there is an
easy trick to implement "or" in a generic way. However, for ranges
there is no such generic trick.

I will check your "range need not be recursive" argument. It
certainly looks reasonable. And I will try to reply in the right
place!

Thanks.

quasi
.

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