Positive polynomial in two real variables

[gvk]

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

 

[Hurkyl]

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?

 

[gvk]

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?

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.

 

[Hurkyl]

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)