Timeline for Polynomials that are sums of squares
Current License: CC BY-SA 4.0
8 events
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| Nov 16, 2020 at 13:27 | history | edited | David E Speyer | CC BY-SA 4.0 |
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| Nov 1, 2009 at 15:49 | comment | added | Jose Capco | Thankks.. yes, I totally missed the 2d part.. I was thinking homogenous in general, if you have even degree of course dehomogenizing gives us a positive polynomial .. and if you have a degree 2d non-homegenous positive polynoomial.. homogenizing it also makes it positive (because of the even degree). I think the "intuition" i got why it feels more likely to have psd than sos, was from 2-variables as you have said things look different as variables increases.. for two variables it took some time after Hilbert's 17th problem that someone could give an example of a positive non-sos polynomial | |
| Nov 1, 2009 at 13:03 | comment | added | David E Speyer | I added a footnote to address your question about homogenous versus nonhomogenous polynomials. | |
| Nov 1, 2009 at 13:03 | history | edited | David E Speyer | CC BY-SA 2.5 |
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| Nov 1, 2009 at 12:59 | comment | added | David E Speyer | Nothing to apologize for. It looks to me like the difference can be summarized in this way up in this way: If you hold n fixed and send d to infinity, then "most" positive polynomials are sums of squares. If you hold d fixed and send n to infinity then "most" are not. Of course, in both cases, you also have to specify the norm being used on the vector space of polynomials. | |
| Nov 1, 2009 at 7:21 | comment | added | Jose Capco | Here is a paper by Netzer and Lassere: tinyurl.com/y8unahb Could anyone teach me how to add hyperlinks when commenting? Anyway, in the paper they used l1-norm on the coefficient.. and from SOS being dense in PSD using that norm i concluded my remark. But to say "it is dense" is definitely not the same as "it is more", sorry for that. | |
| Nov 1, 2009 at 6:36 | comment | added | Jose Capco | I agree, to say "more unlikely" might need a bit more explanation. I will have to find my source and post it. Thanks for pointing out the papers of Blekherman. What I found a bit disturbing was that he used space of homogenous polynomials and made a conclusion over them,but the title and the work seems to confuse me. Is it known that if you use the density computed by Blekhherman for HOMOGENOUS positive polynomials one gets something of the same range for ANY positive polynomial? Also his paper on convex poly, does admit that there are no known non SOS polynomials that are convex | |
| Nov 1, 2009 at 4:01 | history | answered | David E Speyer | CC BY-SA 2.5 |