Timeline for Irreducibility of a resultant of real and imaginary parts of a characteristic polynomial
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Feb 3, 2014 at 23:43 | vote | accept | probably | ||
| Feb 3, 2014 at 23:28 | answer | added | Jason Starr | timeline score: 6 | |
| Feb 3, 2014 at 21:20 | comment | added | probably | Yes, A_i and p_i are real-valued | |
| Feb 3, 2014 at 21:15 | history | edited | probably | CC BY-SA 3.0 |
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| Feb 3, 2014 at 21:13 | comment | added | Jason Starr | Okay, that must mean that you are assuming that all of the matrices $A_i$ and the variables $p_i$ are real-valued. That is what I wanted to know. | |
| Feb 3, 2014 at 21:06 | comment | added | probably | Let's we take a characteristic polynomial of A. This is a polynomial on $\lambda$: $(-1)^n\lambda^n+(-1)^{n-1}tr(A)\lambda^{n-1}+\ldots-\chi_{n-1}(A)\lambda+det(A).$ Thus $R= det(A) + (i\sqrt{\omega})^2\chi_2(A) +\ldots = det A -\omega\chi_2(A) + \omega^2\chi_4(A) - \ldots$ $I= \sqrt{\omega}^{-1}(-\sqrt{\omega}\chi_{n-1}(A)+\omega\sqrt{\omega}\chi_{n-3}(A)+\ldots)=-\chi_{n-1}(A)+\chi_{n-3}(A)\omega+\ldots)$ Therefore $R,I$ are polynomials in $\omega$ and $a_{i,jk},p_i$. We can take a resultant of $R,$ $I$ in $\omega$. It will be a polynomial in $(n^2+1)(k+1)$ variables $a_{i,jk},p_i.$ | |
| Feb 3, 2014 at 19:40 | comment | added | Jason Starr | I am not sure this is a well-posed problem. Can you please explain how $R$ and $I$ are polynomials in $\omega$? Are you assuming some additional hypothesis in $A$? My suspicion is that, once properly formulated, the "generic" version of this question is a simple exercise using "incidence correspondences". However, I have trouble guessing what is really going on. | |
| Feb 3, 2014 at 17:41 | history | edited | probably | CC BY-SA 3.0 |
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| Feb 3, 2014 at 16:41 | history | asked | probably | CC BY-SA 3.0 |