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Positive polynomial in two real variables

  1. Aug 26, 2004 #1

    gvk

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    Does every positive polynomial in two real variables attain its lower bound in the plane? :yuck:
     
  2. jcsd
  3. Aug 26, 2004 #2

    Hurkyl

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    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?
     
  4. Aug 27, 2004 #3

    gvk

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    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.
     
  5. Aug 27, 2004 #4

    Hurkyl

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    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)
     
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