I'm looking for an algorithm to solve a system of $n$ positive integer polynomial equations, assuming that there exists one and only one solution to the system.
Could anyone point me in the right direction or literature?
Example: Assume that the number of variables is $n$, usually we expect $10 \le n \le 50$. Assume that $n=5$ for the example. We need to solve: $$ x_1 = 5 $$ $$ x_1^2 x_2 x_3 = 100 $$ $$ x_2^2 x_3 + x_1 + x_4^2 = 62$$ $$ x_5 x_4 + 2 x_5 = 9 $$ $$ x_1 + x_5 x_2 + x_3 x_4 = 21 $$ $$ x_1 + x_2 + x_3 + x_4 + x_5 = 17 $$ Solution: From $x_1 = 5$ we get that $x_2 x_3 = 4$, thus $(x_2,x_3)=(1,4),(2,2),(4,1)$. From the third equation $x_5(x_4 + 2) = 9$, thus $(x_4, x_5) = (1, 3), (7, 1)$. From the last equation we have that $x_2 + x_3 + x_4 + x_5 = 12$, since the sum of $x_2 + x_3 = 4, 5$ and $x_4 + x_5 = 4, 8$, it means only option is $x_4 = 7, x_5 = 1, x_2 = 2, x_3 = 2$.
Note that I need an algorithm to do this , such that it is automated.