I have a $K$ equations of the form $x\_1^{a\_{i1}} \cdots x\_n^{a\_{in}}=c\_i$ where $a\_{ij}$ are non-negative integer constants and $c\_i$ are real constants -- i.e. each equation is a monomial in $n$ variables and I have $k$ equations. I wish to find all real solutions for $x\_1 \cdots x\_n$. It is assumed that there are a finite number of real solutions. 


First: is there a name for this type of problem?


Second: are there algorithms to solve this problem? Obviously I could use any polynomial system solver but I'm interested in whether there are special algorithms for this problem.



Real solutions for systems of monomial equations
As @Oleg Eroshkin has already pointed out in the comments, this is closely related to solving a linear algebraic system. You could take absolute values and then logs to obtain a linear system of the form $Ax=b$ where the entries of $x$ are absolute values of your original variables. Once you solve that, you're left with an integer programming problem for the signs.


Solving the problem in this way will also tell you qualitatively about the solution set: if $A$ is full rank, then all the solutions are identical except for changes in the signs.


Let's assume $A$ is full rank. Then equivalently, you could just apply "Gaussian elimination" directly to your system in its current form. That is, solve the first equation for $x\_1$:


$$x\_1 = \left(c\_1 x\_2^{a\_{i2}} \cdots x\_n^{a\_{in}}\right)^{1/a\_{i1}}.$$


Use this to eliminate $x\_1$ from the remaining equations. Continuing in this way, you will eventually obtain an "upper triangular" system that you can solve by back substitution. You may sometimes be able to choose certain signs arbitrarily, which will again yield multiple answers.


