# Extrema in two variables of a sum of logs, or equation with sum of rational functions

I am trying to find numerically $\arg\min_{x\in(1, +\infty),y\in(0, 1)}\sum_i\log(xy+\alpha_ix+\beta_iy+\gamma_i)$, where the sum has a large number of terms, and the coefficients are such that the expression is always defined ($\alpha_i, \gamma_i$>0, $\beta_i>-1$).

Of course, one can rephrase this as trying to solve the following system: $\sum_i\frac{y+\alpha_i}{xy+\alpha_ix+\beta_iy+\gamma_i}=\sum_i\frac{x+\beta_iy}{xy+\alpha_ix+\beta_iy+\gamma_i}=0$.

Is there any theory behind this kind of equations?

Thanks in advance.

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Also posted to MO, mathoverflow.net/questions/90323/…. That's very bad form. – Gerry Myerson Mar 6 '12 at 4:20
Sorry, I did not intend to be rude. Is this explicitly discouraged somewhere? – antony Mar 6 '12 at 8:26
It is common sense that if you ask people for help you level with them. Not telling them you're also asking for help somewhere else is not leveling. – Gerry Myerson Mar 6 '12 at 9:05
Further to Gerry's comments: It could easily happen that if Gerry hadn't noticed the cross-post, people on both sites would have put their time into answering the question, not knowing it's already been answered on the other site. – joriki Mar 6 '12 at 11:08
Or I could close the question myself on one site when it is answered on the other site... – antony Mar 6 '12 at 22:41