Timeline for How many products specify a sum?
Current License: CC BY-SA 2.5
4 events
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| Oct 4, 2010 at 2:21 | comment | added | Theo Johnson-Freyd |
So, the missing last step of the answer is the following observation. The $k$ equations define a $(n-k)$-dimensional hyperplane in log-space. But the surface $\sum e^{\log|x_i|}=C$ does not include any hyperplanes (of dimension at least $1$) in the $\log|x_i|$s. So the hyperplane of solutions to the known equations does not pick out a given $C$.
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| Oct 3, 2010 at 22:49 | comment | added | dvitek | It's neither, really, but it's true. A product of some of the $x_i$ becomes a sum of some of the $\log{|x_i|}$, and from here it is basic linear algebra to show that $k \ge n$. | |
| Oct 3, 2010 at 22:39 | comment | added | ronaf | @qiaochu: is your statement a theorem or part of the folklore? | |
| Oct 3, 2010 at 21:51 | history | answered | Qiaochu Yuan | CC BY-SA 2.5 |