Timeline for On strict positivity and Schmüdgen's Positivstellensatz
Current License: CC BY-SA 4.0
14 events
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| Nov 30, 2022 at 20:09 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Aug 2, 2022 at 20:02 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jul 3, 2022 at 19:07 | answer | added | Jens Franke | timeline score: 2 | |
| Mar 21, 2020 at 10:59 | comment | added | SophiaA | @JohannesHuisman Thank you for the article, but what is an example on which the Positivstellensatz does not hold? Thanks a lot. | |
| Mar 17, 2020 at 11:13 | comment | added | user44143 | The paper above is at math.uni-konstanz.de/~scheider/preprints/GUIDE.pdf | |
| Mar 16, 2020 at 21:25 | comment | added | Johannes Huisman | I recommend reading the article C. Scheiderer: Positivity and sums of squares: a guide to recent results. In Emerging applications of algebraic geometry, Volume 149 of IMA Vol. Math. Appl., pages 271–324. Springer, New York, 2009. Section 3 seems to contain results and counter-examples you are looking for. | |
| Mar 16, 2020 at 16:02 | comment | added | user44143 | I gave the following answer, and Emil Jerabek pointed out my error: Consider the set $[-1,1]$ defined by $1+x\ge0$ and $1-x\ge0$. The polynomial $x^2$ is non-negative on the whole set. Any polynomial in $1+x$ and $1-x$ with positive coefficients will have a positive constant term. So there is no way to write $x^2$ in that polynomial form. But, as Emil pointed out, the coefficients only need to be sum-of-squares in the polynomial ring, and not in the base field, so the conclusion of the Positivstellensatz is satisfied here by $$x^2=\left(\frac{x}{\sqrt{2}}\right)^2((1+x)+(1-x))$$ | |
| S Mar 16, 2020 at 13:00 | history | edited | Igor Khavkine | CC BY-SA 4.0 |
added tags, link.
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| S Mar 16, 2020 at 13:00 | history | suggested | Rodrigo de Azevedo | CC BY-SA 4.0 |
added tags.
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| Mar 16, 2020 at 12:37 | review | Suggested edits | |||
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| Mar 16, 2020 at 11:52 | history | edited | YCor | CC BY-SA 4.0 |
fixed English, expanded title
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| Mar 16, 2020 at 11:45 | history | edited | SophiaA | CC BY-SA 4.0 |
added 4 characters in body
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| Mar 16, 2020 at 11:20 | review | First posts | |||
| Mar 16, 2020 at 13:05 | |||||
| Mar 16, 2020 at 11:19 | history | asked | SophiaA | CC BY-SA 4.0 |