Timeline for Complex semi-algebraic sets
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Aug 11, 2019 at 6:48 | comment | added | Dima Pasechnik | I added a discussion of this to the answer. There are conditions on matrices most naturally described in terms of real and imaginary parts, e.g. matrix stability. | |
| Aug 11, 2019 at 6:46 | history | edited | Dima Pasechnik | CC BY-SA 4.0 |
a discussion added.
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| Aug 11, 2019 at 2:20 | review | Low quality posts | |||
| Aug 11, 2019 at 2:55 | |||||
| Aug 10, 2019 at 22:27 | comment | added | Pietro Paparella | I do not disagree here, but the known necessary conditions are given in term of the list of numbers (real or complex) which is more elegant than having to consider the real and imaginary parts (see, e.g., arxiv.org/pdf/1712.05454.pdf). | |
| Aug 10, 2019 at 22:19 | comment | added | Dima Pasechnik | NIEP is, formally, "given $\Lambda$, is there $A\geq 0$ such that...". From this angle, semialgebraic approach gives you an answer. Again, it's perfectly meaningful to ask your question about $|\cdot |$, I do not object to this. | |
| Aug 10, 2019 at 21:21 | comment | added | Pietro Paparella | let me try this one more time: what do the polynomial inequalities in $\Re x_k, \Im x_k$, $1 \le k \le n$, say about the the complex variables $x_1,\dots,x_n$? I can tell you from my experience working on the NIEP that the latter is preferred over the former. | |
| Aug 10, 2019 at 21:20 | comment | added | Dima Pasechnik | I don't understand why you don't consider this appoach a solution to NIEP. For a fixed $n$, it's a question of running a finite exact algorithm on the system (4) of polynomial equations and inequalities with integer coefficients to obtain $S$, and thus a description of the $\mathbb{L}^n$. | |
| Aug 10, 2019 at 21:06 | comment | added | Dima Pasechnik | @PietroPaparella in your comment to your question you said "it may be possible that their conclusion is correct, but their argument that leads to the conclusion is not. ". I have trouble matching "I fully understand" with "it may be possible that...", which clearly says you had doubts about their argument. I am glad my answer helped to clarify this. | |
| Aug 10, 2019 at 21:05 | comment | added | Pietro Paparella | I respectfully disagree. Their treatment of that topic is slipshod (although somewhat acceptable because they were trying to summarize). The inequalities that are required for the solution of the NIEP are in terms of the complex variables, not the real and imaginary variables, which is why my question is pertinent. | |
| Aug 10, 2019 at 21:03 | comment | added | Dima Pasechnik | They never say in [MR2399570] explicitly what variables they use. While your question on whether $|\cdot |$ would suffice to produce the needed inequalites is fine, there is nothing wrong with [MR2399570]. | |
| Aug 10, 2019 at 19:34 | comment | added | Pietro Paparella | What was claimed is that the set is semi-algebraic in the complex variables (which is non-sensical) not the real variables. Again, I fully understand that it is semi-algebraic in the real variables but you continue to not understand my objection. | |
| Aug 10, 2019 at 19:31 | history | answered | Dima Pasechnik | CC BY-SA 4.0 |