Consider system of non-linear equations $$ \begin{align} \ln \frac{x_1}{1 - x_1 - \ldots - x_n} + P_1(x_1, \ldots, x_n) &= 0 \\ &\ldots \\ \ln \frac{x_n}{1 - x_1 - \ldots - x_n} + P_n(x_1, \ldots, x_n) &= 0 \end{align} $$ with $P_i(x_1, \ldots, x_n)$ being polynomials with coefficients in $\mathbb{R}$ and $x_1, \ldots, x_n$ confined to the interior of the standard real symplex $x_1, \ldots, x_n \in \mathbb{R}$, $x_1, \ldots, x_n > 0$, $x_1 + \ldots + x_n < 1$.

Is there a general method for solving such a system analytically and/or numerically?

Can we compute its Groebner basis using standard algorithms for systems of polynomial equations and determine if it's positive- or zero-dimentional? In what ring we would compute it? $\mathbb{R}[x_1, \ldots, x_n, l_1, \ldots, l_n]$ with $$ l_i = \ln \frac{x_i}{1 - x_1 - \ldots - x_n}? $$

If the system proves to be zero-dimentional, can we somehow bound its number of solutions from above (perhaps using some generalization of Bézout's theorem?), so that we may be able to solve it numerically via homotopy continuation starting from some system of polynomial equations with suitable number of solutions?


1 Answer 1


We could consider this system as a system of $n$ complex-analytic equations in $n$ variables $x_1, \ldots, x_n \in \mathbb{C}^n$.

If we choose a suitable region $D \subset \mathbb{C}^n$ such that it contains standard real simplex and does not have solutions of our system on its boundary $\partial D$, theorem 2.4 from [1] guarantees that out system would have only a finite number of isolated solutions in the region $D$.

Then we could use numerical methods from chapter 4 "Systems of analytic equations" of [2] to find these solutions, and then choose only those belonging to the standard real simplex.

How can we choose region $D$? Seems that polydisk of radius $r > 1$ with branch cuts along negative real axes would do the job.

[1] Aizenberg, L. A., and A. P. Yuzhakov. Integral Representations and Residues in Multidimensional Complex Analysis. Translations of Mathematical Monographs. Providence, Rhode Island: American Mathematical Society, 1983. https://www.amazon.com/Representations-Multidimensional-Translations-Mathematical-Monographs/dp/0821815504.

[2] Kravanja, Peter, and Marc Van Barel. Computing the Zeros of Analytic Functions. Lecture Notes in Mathematics. Springer-Verlag Berlin Heidelberg, 2000. https://doi.org/10.1007/BFb0103927.


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