Positive polynomial in two real variables

Click For Summary

Discussion Overview

The discussion revolves around whether every positive polynomial in two real variables attains its lower bound in the plane. Participants explore theoretical implications and properties of continuous functions in relation to this question.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions if every positive polynomial in two variables attains its lower bound.
  • Another participant suggests investigating how a continuous function might fail to attain its lower bound, hinting at potential counterexamples.
  • A further contribution mentions Sylvester's theorem, proposing that a positive definite polynomial never reaches the plane, implying a connection to the original question.
  • Another participant proposes examining the properties of continuous functions that do not attain their lower bounds and suggests proving that polynomials cannot have those properties.
  • There is a request for clarification on how Sylvester's theorem would be applied in this context, indicating some uncertainty about its relevance.

Areas of Agreement / Disagreement

Participants express differing views on the implications of continuous functions and the application of Sylvester's theorem, indicating that multiple competing perspectives remain unresolved.

Contextual Notes

There are unresolved assumptions regarding the properties of continuous functions and the specific conditions under which a polynomial might fail to attain its lower bound. The relevance of Sylvester's theorem is also not fully clarified.

gvk
Messages
83
Reaction score
0
Does every positive polynomial in two real variables attain its lower bound in the plane?
 
Physics news on Phys.org
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?
 
Hurkyl said:
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.
 
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)
 

Similar threads

MHB Real Roots of Polynomial Minimization Problem
  • · Replies 1 ·
Replies
1
Views
1K
High School Intermediate Value Theorem and Synthetic Division
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Quintic Polynomial Tschirnhaus Transform: Possible complex Bring Radicals
  • · Replies 0 ·
Replies
0
Views
3K
Undergrad Want to understand how to express the derivative as a matrix
  • · Replies 8 ·
Replies
8
Views
3K
Substitute PID Controls with a Polynomial Equation/Table?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Finding the Largest (or smallest) Value of a Function - given some constant symmetric errors
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Approximation of a function of two variables
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad What would a calculus author have to say on ##\int r^2dm##?
  • · Replies 11 ·
Replies
11
Views
4K
Graduate Key difference between two real and single complex variable?
  • · Replies 4 ·
Replies
4
Views
2K
MHB Show that the polynomial has no real roots
  • · Replies 3 ·
Replies
3
Views
3K