I have a similar problem addressed in System of Equations Generator. What I need is an automatic way of generating a system of equations with unique solutions, but the equations are not exclusively linear.
The method of working backwards, by randomly assigning integer values to variables and then generating the coefficients and computing the determinant matrix is perfect for a system of linear equations of the form:
\begin{align}x + y = 7\\3x - 2y = 6\end{align}
I can verify that the solution is unique with x = 4 and y = 3.
Now I would like to add two more operators: multiplication and division. The expected equations would be like this:
\begin{align}x + y = 4\\2x * y = 6\end{align}
How to generate these systems of equations and verify if the solution is unique, assuming integer variables and up to 5 variables? I found a lot of papers about uniqueness in non-linear equations, but I guess my requirement is a little simpler.