Is it possible to write a polinomy in R like a square of "something" if and only if that polinomy(in n variables, and grade m, with n and m naturals, ) is all time positive for all values? if it is possible consider another problem: i, is there a method to write all positive polinomy in R like a square of something?
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1Possible duplicate of Prove that $p \in \mathbb{R}[x]$ can be represented as a sum of squares of polinomials from $\mathbb{R}[x]$ – Martin R Aug 16 '17 at 15:22
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By polinomy do you mean polynomial? I think we usually call it degree of a polynomial rather than grade. – Martin Sleziak Aug 16 '17 at 15:25
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I am not particularly happy with self-invented jargon such as "polinomy" and "all time positive", but maybe that is just me? – Hans Hüttel Aug 16 '17 at 15:25
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@HansHüttel. It appears to be a translation difficulty. – DanielWainfleet Aug 17 '17 at 01:15
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Well you can write $p(x)$ as the square of $\sqrt{p(x)}$.
More interestingly, you can write a real polynomial in one variable that is always non-negative as a finite sum of squares of real polynomials.
You can also write a real polynomial in two or more variables which is always nonnegative for real arguments, as a finite sum of squares of rational functions with real coefficients, but not necessarily the sum of squares of real polynomials.
Angina Seng
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