Re: ranges of integer polynomials

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

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

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.

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. 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!

.

Relevant Pages

  • Re: Possible results from three variables
    ... For positive integers n,k, let fbe the number of possible ... subgroup of R^m, i.e. a lattice. ... Let K be the polytope ... polynomials in k with rational coefficients. ...
    (sci.math)
  • Re: Possible results from three variables
    ... For positive integers n,k, let fbe the number ... since they have combinatorical meaning in a similar way to the original question here. ... and also some kind of Ehrhart polynomials, but ill stick with euler at first sight. ...
    (sci.math)
  • Re: Possible results from three variables
    ... For positive integers n,k, let fbe the number of possible ... fixed polytope. ... A quick search turns up Ehrhart polynomials ... subgroup of R^m, i.e. a lattice. ...
    (sci.math)
  • Re: ranges of integer polynomials
    ... Note that if A is a set of positive integers, ... but have the same recursive ranges ... which of these other, "hyperbolic", polynomials ... This gets at the heart of the undecidability issue. ...
    (sci.math)
  • @quasi : deny "ab + cd"
    ... i was thinking about the set ab + cd with all variables integers>=2. ... the set of positive integers that are the sum of 2 composites. ... suppose we allow polynomials of integer and primevalued variables; then can deny be stated in such a polynomial variant? ...
    (sci.math)