A way of checking that a multivariate system of equations has a solution?
Is there a way to check that a multivariate system of polynomial equations does indeed have a solution(or not)? The context i am using this in is as follows: say i have a system of multivariate polynomial equations over a finite field, i then put values in for some of the variables, and then i need to check if the resulting system in the remaining unassigned variables has a solution or not? If not, then i will reassign different values to the other variables until a system with a solution is yielded. Of course because the assigning of these values is random there is a high chance the system has a solution anyway, but in the event that it does not is where i need a way to check whether a solution exists or not.
There are ways to do that in Sage, but please provide a concrete example, it will be easier for us to show how to !
I concur with @tmonteil -- please provide an example of input of a polynomial system that other users can copy paste to get started exploring your question.