[FOM] Polynomials/correction

Harvey Friedman friedman at math.ohio-state.edu
Tue Jan 24 17:17:14 EST 2006

Joseph Miller pointed out that a side remark I made about polynomials with
real arguments is false.

In http://www.cs.nyu.edu/pipermail/fom/2006-January/009603.html I wrote

> THEOREM A. Every polynomial of several variables, with integer coefficients,
> assumes a value closest to the origin.
> For real arguments, the statement is provable in, say, intuitionistic RCA0.

Joe Miller writes:

Not that it's an important point, but the theorem isn't true for real
arguments. Take for example:

    f(x,y) = (x^2 - y^2 + 1)^2 + (x-y)^2

This polynomial takes strictly positive values but if
    x = sqrt(n) and
    y = sqrt(n+1),
then f(x,y) = (sqrt(n)-sqrt(n+1))^2. So f(x,y) can be made arbitrarily
close to zero by taking n large.

I'm sure this example is well known, but to give credit where it's
due, the phenomenon was pointed out to me -- as a mind teaser -- by
Michael O'Connor, who is a grad student at Cornell.


Harvey Friedman

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