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System of crosspolynomials of variable degree 
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#1
Mar413, 09:17 AM

P: 1

Dear all,
I am stuck with an apparently easy system of 3 simultaneous equations that has to be solved in x,y and z. The system is the following: (y+z)/((x+y+z)²)=ax^{α1} (x+z)/((x+y+z)²)=by^{α1} (x+y)/((x+y+z)²)=cz^{α1} The parameters (a,b,c) and the variables (x,y,z) are all in ℝ₊. The parameter α is in the open interval (0,1), so that the problem is not trivial. The question is for which α there exists a closedform solution, and for which α there exists none, given a general vector (a,b,c). Proving that there always exist a unique solution is trivial as in each equation the righthand sides are always increasing, whereas the lefthand sides are always decreasing in the corresponding variable (x for the first, y for the second, z for the third). Hence, my problem is in the existence of a closedform solution. Many thanks!! 


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