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Are there any known methods which can be used to solve system of equations of the form $$ \begin{align} \ln \frac{P_1(x_1,\ldots,x_n)}{Q_1(x_1,\ldots,x_n)} + R_1(x_1,\ldots,x_n) &= 0 \\ &\ldots \\ \ln \frac{P_n(x_1,\ldots,x_n)}{Q_n(x_1,\ldots,x_n)} + R_n(x_1,\ldots,x_n) &= 0, \end{align} $$ where $P_i(x_1,\ldots,x_n)$, $Q_i(x_1,\ldots,x_n)$ and $R_i(x_1,\ldots,x_n)$ are polynomials, analytically and/or numerically in the positive reals $x_i \in \mathbb{R}_+$?

Perhaps at least in the special case when $P_i(x_1,\ldots,x_n)$ and $Q_i(x_1,\ldots,x_n)$ are linear and solutions are confined to the interior of the standard simplex $x_1 + \ldots + x_n < 1$, $x_i > 0$?

As far as I know, systems of polynomial equations can be solved analytically using Groebner bases, and numerically using homotopy continuation method. Can these methods be adapted for system of equations shown above?

Additional tag: homotopy-continuation

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  • $\begingroup$ what are the variables? $\endgroup$ Dec 28 '17 at 10:00
  • $\begingroup$ can you post us an example of such a System? $\endgroup$ Dec 28 '17 at 10:05
  • $\begingroup$ @MartínVacasVignolo Variables are $x_1, \ldots, x_n$. $\endgroup$
    – myrix
    Dec 28 '17 at 10:19
  • $\begingroup$ @Dr.SonnhardGraubner $$ \begin{align} \ln \frac{x_1}{1 - x_1 - x_2} - \frac{d}{dx_1} \Bigg[ \sum_{k_1, k_2} C_k^{k_1, k_2} x_1^{k_1} x_2^{k_2} (1 - x_1 - x_2)^{k - k_1 - k_2}& \cdot \\ \big[ x_1 \ln a_{1, k_1, k_2} + x_2 \ln a_{2, k_1, k_2} + (1 - x_1 - x_2) \ln a_{3, k_1, k_2} \big] \Bigg] &= 0 \\ \ln \frac{x_2}{1 - x_1 - x_2} - \frac{d}{dx_2} \Bigg[ \sum_{k_1, k_2} C_k^{k_1, k_2} x_1^{k_1} x_2^{k_2} (1 - x_1 - x_2)^{k - k_1 - k_2}& \cdot \\ \big[ x_1 \ln a_{1, k_1, k_2} + x_2 \ln a_{2, k_1, k_2} + (1 - x_1 - x_2) \ln a_{3, k_1, k_2} \big] \Bigg] &= 0 \end{align} $$ $\endgroup$
    – myrix
    Dec 28 '17 at 10:30
  • $\begingroup$ @Dr.SonnhardGraubner E.g. when we want to find critical points of expectation of Kullback-Leibler divergence $D(x_1, x_2,1 - x_1 - x_2; a_{1, k_1, k_2}, a_{2, k_1, k_2}, a_{3, k_1, k_2})$ over all possible multinomial k-samples $k_1, k_2, k - k_1 -k_2$, $0 \le k_1, k_2 \le k$ parametrized by a set of distributions $\{ a_{(1, 2, 3), k_1, k_2} \} \in \mathbb{R}^{3 C_{k+2}^2}$. $\endgroup$
    – myrix
    Dec 28 '17 at 10:37
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Upon further investigation it appears that, at least, any such system can be considered as a system of complex-analytic equations, which can be solved in a suitably chosen region of $\mathbb{C}^n$ containing standard real simplex numerically via methods from the chapter chapter 4 "Systems of analytic equations" of the book "Computing the Zeros of Analytic Functions" by Peter Kravanja and Marc Van Barel.

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