Timeline for Simultaneous Equations Involving Power Sums
Current License: CC BY-SA 2.5
25 events
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| Nov 24, 2009 at 1:41 | vote | accept | Kaveh Khodjasteh | ||
| Nov 23, 2009 at 19:06 | history | edited | Kaveh Khodjasteh | CC BY-SA 2.5 |
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| Nov 23, 2009 at 18:57 | history | edited | Kaveh Khodjasteh | CC BY-SA 2.5 |
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| Nov 23, 2009 at 2:20 | history | edited | Greg Kuperberg |
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| Nov 23, 2009 at 2:04 | answer | added | fedja | timeline score: 11 | |
| Nov 18, 2009 at 15:38 | comment | added | Kaveh Khodjasteh | Notice that one could set one of the $x_i$ or $y_i$ arbitrarily equal to one and simplify the setting to a question about $n$ pairs of positive numbers. Let me be more explicit: There exists $c,C_1,C_2$, all positive, such that $n$ pairs $(x_i,y_i)$ can be found such that $$\sum_{i=1}^n x_i^k-\sum_{i=1}^{n} y_i^k=1$$ with $k=\ell,\cdots,2\ell-1$. We also need the variables $x_i,y_i$ to be all between $C_1$ and $C_2$ and that $n\le c\ell$. | |
| Nov 18, 2009 at 15:38 | history | edited | Kaveh Khodjasteh | CC BY-SA 2.5 |
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| Nov 18, 2009 at 15:14 | history | edited | Kaveh Khodjasteh | CC BY-SA 2.5 |
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| Nov 18, 2009 at 4:41 | answer | added | David E Speyer | timeline score: 4 | |
| Nov 17, 2009 at 17:17 | history | edited | Kaveh Khodjasteh | CC BY-SA 2.5 |
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| Nov 17, 2009 at 15:09 | history | edited | Kaveh Khodjasteh | CC BY-SA 2.5 |
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| Nov 17, 2009 at 12:13 | answer | added | David E Speyer | timeline score: 3 | |
| Nov 16, 2009 at 20:10 | vote | accept | Kaveh Khodjasteh | ||
| Nov 16, 2009 at 20:10 | |||||
| Nov 16, 2009 at 18:42 | comment | added | Kaveh Khodjasteh | Thanks Steve for the clarification. Yes, actually I had thought about it but the eigenvalue problem reduces to finding positive roots of polynomials and matching them! I did try a bit along those directions. What Darsh wrote seems to be the right direction though, although lower bounding these 'small enough' numbers is a little troublesome. | |
| Nov 16, 2009 at 18:18 | vote | accept | Kaveh Khodjasteh | ||
| Nov 16, 2009 at 20:09 | |||||
| Nov 16, 2009 at 12:21 | comment | added | Steve Flammia | What Qiaochu is saying is the following. Let P_k be the power sum with exponent k in n variables x_1,...,x_n. Suppose you know P_k for k=1,...,n. Then you can use the Newton-Girard identities to convert these by polynomial combinations into the elementary symmetric polynomials. Once you have the ESPs, you can easily form the characteristic polynomial for the diagonal matrix whose entries are x_i along the diagonal. Thus, the power sums exactly determine the spectrum of this matrix, and this matrix contains your variables as its eigenvalues. | |
| Nov 16, 2009 at 8:54 | history | edited | Kaveh Khodjasteh | CC BY-SA 2.5 |
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| Nov 16, 2009 at 8:37 | comment | added | Kaveh Khodjasteh | Thanks for the comments: I am not sure about n+1 consecutive power sums uniquely determining n+1 numbers. I guess it depends on the number of variables in the sum, right? The number of variables is n+1 on one side and n on the other side. I want to keep all of them neither too small nor too large, preferably between 1 and 2! I think Reid got it right, with his quantifier statement. The number of them will have to scale nicely with the number of equations [the powers l to 2l-1]. | |
| Nov 16, 2009 at 8:01 | answer | added | Darsh Ranjan | timeline score: 4 | |
| Nov 16, 2009 at 7:35 | comment | added | Reid Barton | I guess the question is "Do there exist C, 0 < C_1 < C_2 such that for every l, there exists n < Cl and C_1 < x_i, y_i < C_2 such that ..." | |
| Nov 16, 2009 at 7:22 | comment | added | Qiaochu Yuan | I also don't understand which variables you want to make large and which you want to keep fixed, nor do I understand the extent to which you don't want x_i = y_i, y_{n+1} = 0 to be a valid solution. | |
| Nov 16, 2009 at 7:08 | comment | added | Qiaochu Yuan | Are you aware that n+1 consecutive power sums uniquely determine n+1 numbers up to permutation? | |
| Nov 16, 2009 at 6:42 | comment | added | Kim Morrison | Can you provide some motivation? | |
| Nov 16, 2009 at 6:41 | history | edited | Kim Morrison | CC BY-SA 2.5 |
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| Nov 16, 2009 at 6:16 | history | asked | Kaveh Khodjasteh | CC BY-SA 2.5 |