Timeline for When does 'positive' imply 'sum of squares'?
Current License: CC BY-SA 4.0
7 events
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| Oct 1, 2018 at 15:35 | comment | added | Ryan O'Donnell | Oops, hit 'enter' before I was finished typing the comment. Their example: $p(x,y) = x^2 y^2(x^2+y^2-1) + 1$ is nonnegative on $\mathbb{R}^2$, but is not the sum of squares of polynomials. Nonnegativity is almost trivial: certainly a nonnegative point would have to have $x^2 + y^2 \leq 1$; but then $x^2y^2 \leq 1$ so $x^2 y^2(x^2+y^2-1) \geq -1$. The proof that $p$ cannot be written as $q_1^2 + \cdots + q_n^2$ is also very brief; see the paper for the four-sentence elementary argument. | |
| Oct 1, 2018 at 15:29 | comment | added | Ryan O'Donnell | Better than Motzkin, I like the example by Berg-Christensen-Jensen: | |
| Oct 1, 2018 at 12:11 | history | edited | Wolfgang | CC BY-SA 4.0 |
corrected typo in formula
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| Oct 1, 2018 at 12:06 | history | edited | Wolfgang | CC BY-SA 4.0 |
corrected typo in formula
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| May 10, 2011 at 11:16 | comment | added | Emil Jeřábek |
In any field, nonzero sums of squares form a multiplicative group. (Pfister has shown that the same holds for sums of $2^n$ squares for any natural number $n$.) In particular, inverses of sums of squares are themselves sums of squares, since $(\sum_ia_i^2)^{-1}=\sum_i(a_i/a)^2$, where $a=\sum_ia_i^2$.
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| May 10, 2011 at 1:33 | comment | added | Ryan Reich | Is it true, then, that a positive quadratic form $f(x,y) \in \mathbb{Z}[x,y]$ can be written as a sum of squares in the fraction field $\mathbb{Q}(x,y)$? One could imagine obvious exceptions where the denominator is a sum of squares, so perhaps the correct question is: can any positive function in $\mathbb{Q}(x,y)$ be written as a sum of squares and inverses of sums of squares? | |
| Jan 10, 2011 at 10:45 | history | answered | Tony Carbery | CC BY-SA 2.5 |