Re: ranges of integer polynomials

On Oct 5, 9:57 pm, quasi <qu...@xxxxxxxx> wrote:

problem (8):

If range(f) is a subset of N, must range(f) be recursive?

how are these steps?

.. if the range is only positive
then the polynomial must be of even degree in each variable
(fixing all other variables demonstrates)

.. for any polynomial f of even degree in each variable
given any n e N
there are only a finite number of points
suchThat f(x1, x2, ..., xn) <= n

.. let dmin be the minimum degree of f over all variables
thereExists a C e R+ suchThat
all points X with f(X) <=n
have x1^dmin + x2^dmin + ... + xn^dmin < C n
which is a well defined subcollection S(n) of Z^n

.. all you need to do
to determine if any value n is in the range
is to check the finite number of points S(n)

this gives a yes/no answer

and so range(f) is recursive

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galathaea: prankster, fablist, magician, liar

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