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  Wrong equation in a paper about cosmic topology

+ 4 like - 0 dislike

I am studying the paper  " Exact Polynomial Eigenmodes for Homogeneous Spherical 3-Manifolds"  (https://arxiv.org/pdf/math/0502566v3.pdf)

I am trying to obtain the polynomials given by the equation (43) on page 19.  I am noting that there is an error in the first and the third polynomials in equation (43).   According with the mentioned paper:

but the correct expressions are

Could you confirm the error and the correct expressions?  Many thanks

asked Oct 13, 2016 in Mathematics by juancho (1,130 points) [ no revision ]

1 Answer

+ 4 like - 0 dislike

$T_8 = T'_{4a}T'_{4b} = ({{{\alpha}^{4}}+{2 i {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}) ({{{\alpha}^{4}}-{2 i {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}) =\\
{{{\alpha}^{8}}+{6 {\alpha}^{4} {\beta}^{4}}}+{{\beta}^{8}}$


$T_{12} = \frac {T_{4a}^{'3} + T_{4b}^{'3}}{2} = \frac{{{{{({\alpha}^{4}}+{2 i {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}})^{3}}+{{{{({\alpha}^{4}}-{2 i {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}} )^ {3}}}{2} = \\
{{{{\alpha}^{12}}-{9 {\alpha}^{8} {\beta}^{4}}}-{9 {\alpha}^{4} {\beta}^{8}}}+{{\beta}^{12}}$

while with the missing $\sqrt(3)$ , we get correctly (44) :

$T_8 = T'_{4a}T'_{4b} = ({{{\alpha}^{4}}+{2 i \sqrt{3} {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}) ({{{\alpha}^{4}}-{2 i \sqrt{3} {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}) =\\
{{{\alpha}^{8}}+{14 {\alpha}^{4} {\beta}^{4}}}+{{\beta}^{8}}$


$T_{12} = \frac {T_{4\alpha}^{'3} + T_{4\beta}^{'3}}{2} = \frac{{{{{({\alpha}^{4}}+{2 i \sqrt{3} {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}})^{3}}+ {{{{({\alpha}^{4}}-{2 i \sqrt{3} {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}} )^ {3}}}{2} = \\
{{{{\alpha}^{12}}-{33 {\alpha}^{8} {\beta}^{4}}}-{33 {\alpha}^{4} {\beta}^{8}}}+{{\beta}^{12}}$

Idem for the following. I mean the author didn't used the equation with the typo error for the next calculus.

Then it is probably just a well found typo error :) You must write to the author to release a new version.

answered Oct 13, 2016 by igael (360 points) [ revision history ]
edited Oct 13, 2016 by igael

@igael, you are very right.  Many thanks.  All the best.

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