Re: density of the range of an integer polynomial

On Fri, 05 Oct 2007 03:40:39 GMT, Gerry Myerson
<gerry@xxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

In article <r18bg3hem590879ct2jidonu3n4p41h2hp@xxxxxxx>,
quasi <quasi@xxxxxxxx> wrote:

On Tue, 02 Oct 2007 23:13:13 -0500, Robert Israel
<israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

(2) For an integer polynomial, what densities are possible? Is every
rational number in the interval [0,1] achievable as a density? Can a
density be irrational?

For density m/n, try f(x,y) = x + n y + (x^9 + y^9) prod_{j=1}^m (x - j).

I can verify the correctness of the above experimentally, but I don't
see why it works.

It's pretty clear you get all the numbers congruent 1, 2, ..., m mod n
by taking x = 1, 2, ..., m and y running through Z. If x is anything
else then the y^9 term kicks in and gives terms with zero density.

For the irrational density 1 - 6/pi^2, try
f(v,w,x,y,z) = v (w^4 + x^4 + y^4 + z^4 + 2)^2

For this one, I'll guess that it's related to Buffon's Needle --
either that or there's an algebraic tie to an infinite series, but I
don't have a clear strategy. Moreover, it's not even that easy to
verify experimentally.

The polynomial generates the integers that are not squarefree,
that is, it generates those positive integers that have a square
factor exceeding 1. It known (see any good intro number theory
text) that the squarefree numbers have density pi^2 / 6.

Thanks -- mystery resolved.

quasi
.

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