When polynomials go wild

Every fall I took an advanced problems seminar starting with my 2nd year of undergrad. In that course the professor took problems or variations of problems from the Putnam Competition to prepare a team for the university.

This problem is not from the exam, but is related to an exam question and was presented to us in this course.

A polynomial in a single real variable is a very well behaved function. It's easy to find its minima and maxima, and you know that it happily charges off to infinity or negative infinity as you give it bigger and bigger positive and negative numbers.

But the problem that was posed to us one year was this - can you create a polynomial in 2 real variables that has the range (0,inf). Note that the range asked for is NOT [0,inf]. You need to create a polynomial in 2 real variables that you can show gets as close to zero as possible, but never attains it.

Although most mathematicians might solve this pretty quickly, I remember tinkering around with the problem for a while playing around with the relationship of the 2 variables to take slices of what would be a surface. As I had lunch with one of my professors I described the problem and told him how I was playing around with it.

I explained that no matter how I sliced the surface I didn't seem to be able to create an equation that had the desired behavior. I described how I made the slices of the surface - by substituting a linear relationship between the two variables when it hit us both at the same time.

By looking at slices like this, I was always reducing the problem to a polynomial in 1 real variable which would NEVER exhibit the behavior I was looking for. We both knew it was the key to the problem.

I started playing around with the a relationship between the two variables, and not long after that I had created my function and was able to prove it had the range (0,inf).

I'll update this entry later with one possible answer, but I'll let you play around with the idea yourself for a while.

Enjoy!