# Positive polynomial in two real variables

1. Aug 26, 2004

### gvk

Does every positive polynomial in two real variables attain its lower bound in the plane? :yuck:

2. Aug 26, 2004

### Hurkyl

Staff Emeritus
Let's start by investigating how it could fail.

Do you know of any way that a continuous function can fail to attain its lower bound?

3. Aug 27, 2004

### gvk

Do you mean the function which asymptoticaly aproaches the plane when x ->infinity?
It seems to me that, according the Sylvester's theorem the positive defined polynomial never reaches the plane, and it does not matter how behave the continuous function.

4. Aug 27, 2004

### Hurkyl

Staff Emeritus
I was suggesting a possible line of attack: examime what properties a continuous function must have if it doesn't attain its lower bound, then prove a polynomial can't have those polynomials.

But it sounds like you already have a line of attack... how are you proposing to use Sylvester's theorem?

(I don't recall the theorem; a quick google search doesn't provide anything that seems relevant)