Is this Polynomial an SOS Polynomial?

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Discussion Overview

The discussion centers around the characterization of a specific polynomial, F, and whether it qualifies as a sum of squares (SOS) polynomial. The scope includes theoretical considerations of polynomial properties and their implications in mathematical analysis.

Discussion Character

  • Debate/contested, Technical explanation

Main Points Raised

  • One participant notes that the polynomial F approaches negative infinity as x approaches positive or negative infinity, suggesting that it cannot be a sum of squares since such sums are always nonnegative.
  • Another participant seeks clarification on the nature of the polynomial, asking for its classification and expressing uncertainty about the terminology used (function vs. equation).
  • The same participant characterizes the polynomial as a degree six polynomial and suggests it may not be suitable for analysis through traditional analytic methods.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the classification of the polynomial as an SOS polynomial, with one viewpoint asserting it cannot be, while others seek further clarification on its nature.

Contextual Notes

There are unresolved assumptions regarding the properties of the polynomial and its behavior under various conditions, particularly concerning its classification as a sum of squares.

karate
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F = -0.2662*x^6 + 48.19*x^5 - 3424.2*x^4 + 121708*x^3 - 2*e^6*x^2 + 2*e^7*x - 6*e^7;
 
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As mentioned in your other thread, ##\lim_{x\rightarrow\pm\infty}F(x)=-\infty##. So ##F## is sometimes (you could even say way more often than not) negative. Since a sum of squares is always nonnegative, your polynomial is not a sum of squares.
 
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thank you sir, so what do you think is this type of equation?
 
karate said:
thank you sir, so what do you think is this type of equation?

Well, first off the thing that (I think) you're asking me about is a function, not an equation. Secondly, I would characterize it as a polynomial of degree six.

Edit: I would also characterize this polynomial as apparently not being particularly well-suited for examination by analytic means.
 
Last edited:

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