Re: density of the range of an integer polynomial

quasi <quasi@xxxxxxxx> writes:

For an integer polynomial f, possibly multivariate, let range(f)
denote the range of f for all possible integer inputs, and let
density(f) denote the density of (range(f) intersect N), as a subset
of N.

Note -- as discussed in a recent thread, density(f) is not necessarily
the same as density(-f).

Of course, density(f), if it exists, is a real number in the interval
[0,1].

Some simple examples ...

If f(x) = x^2, then density(f) = 0.

If f(x) = k*x, where k is a fixed positive integer, then density(f) =
1/k.

Two questions ...

(1) Does every integer polynomial have a density?

(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).

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
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

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