Proof of a Positive and Infinitely Small Polynomial

Date: 10 Apr 1995 09:53:43 -0400
From: Marek Kacprzak
Subject: Polynomials

Dear Mathematical Society!
I have a math problem!  If You are able to help me I will be very
grateful to You.  And the Problem is following:

Prove ,that there exists a two variables polynomial W(x,y) such that
for any x and y  it is always positive but at the same time 
infinitely small.

Please help me to solve it.                

                                                  Marek Kacprzak

Date: 10 Apr 1995 19:20:09 -0400
From: Dr. Ken
Subject: Re: Polynomials

Hello there!

Well, I'm not sure I understand the problem.  The problem is that there is
no number that is infinitely small and still positive.  Perhaps something is
lost in translation?  If you can clarify the problem any more, please write
back.

-Ken "Dr." Math

Date: 11 Apr 1995 05:43:08 -0400
From: Marek Kacprzak
Subject: Re: Polynomials

Dear Dr Williams!
Perhaps I have written my problem wrong. I don't know:
Does this polynomial really exist?
This is the question.  And now I give the problem once again:

Does there exist a two variables polynomial W(x,y) such that for any x,y
it is always positive (everywhere) W(x,y)>0 and at the same time
infinitely small.Maby it doesnt exist,but it must be checked by
a mathematical way (Proof)

Date: 16 Apr 1995 13:09:21 -0400
From: Dr. Ken
Subject: Re: Polynomials

Hello there!

Well, I don't think that such a polynomial can exist, because what you're
asking is that the polynomial has a nonexistent value for all x and y,
namely a value that is smaller than all positive numbers but greater than
zero; because if a non-negative number is less than all positive numbers,
then it's dead zero.  

You could, however, give a different name for the zero function, which will
_look_ like it's a function that you're looking for, but in fact it's just
the zero function:

F(x,y) = Limit as n goes to Infinity of1/(x^2 + y^2 + 2)^n

-Ken "Dr." Math

Date: 21 Apr 1995 22:39:13 -0400
From: Marek Kacprzak
Subject: Polynomial

Dear Dr Williams!

This Polynomial exists-I am 100% sure,that this polynomial exists.
I need only a Proof!
Please try to help me find a solution! Try to consult with other 
people who can know something more about this type of problem about 
this problem. Once again, it is as follows:

Prove,that exists a two variable Polynomial W(x,y) ( not function)
such that is always positive and at the same time infinitely small.

Date: 21 Apr 1995 22:39:13 -0400
From: Dr. Ken
Subject: Re: Polynomial

Hello.

When you wrote before, you said that you were looking for a proof _or_ 
a disproof of this phenomenon.  How do you know that such a structure 
exists?

Here is some additional material.  The size of _any_ polynomial with
coefficients in the real numbers will tend toward infinity as its input
values tend toward infinity, unless the polynomial is a constant
polynomial.  In that case, the question becomes "is there a constant 
number c such that c is positive and infinitely small?"  Well, that 
depends.

Some people allow infinitesimal numbers in the real numbers (I personally
don't think about them like that, but I suppose it's a personal decision).
An infinitesimal number is one that is positive but less than every
positive real number.  So if you allow polynomials with infinitesimal
coefficients, then sure, use a constant polynomial whose value is
infinitesimal.  But it's still not a very interesting solution.  My thanks
to Professor Grinstead and Ethan Magness for discussing your question with
me.

-Ken "Dr." Math