I made this mostly as a fun challenge :)

You are right that there is some arbitrariness involved when picking a solution, however it's a bit more subtle than that.

Let's say our problem has N free variables.

Step 1 is finding the subset of R^N that is the solution to the root finding problem. If this subset is a point, we are done (return that point). Note that if there is no solution at all bidicalc should correctly report it.

Step 2 is if the solution subset is not a point. Then there is multiple (maybe even an infinity of) solutions, and picking one is indeed arbitrary.