# 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? (an actual way to solve this is welcome but I'd like to know about theory as well.)

What is the domain of definition? What are the signs of the coefficients? DO you really mean arg min, as opposed to argmax (notice that at the origin the sum equals $-\infty,$ so that's a pretty good candidate for arg min...) –  Igor Rivin Mar 6 '12 at 1:21