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I'm here because I lack experience and education in the subject of systems of polynomial equations (univariate and multivariate). I've also recently found a need to work with them. If this is the wrong subforum, then I apologize for my ignorance and would be grateful if a moderator would move it into the correct section.

Now, onto the subject.

I have a system of multivariate polynomial equations. The system is over-determined and I know that at least one solution exists. This solution is 'trivial' (in regards to what I'm doing) and simply consists of all indeterminants, y_j, taking the value 0. I also know that only a finite number of solutions exist (I think this is called zero-dimensional?)

Now, the thing is, I don't really care about this solution. Furthermore,I don't need to find every solutionof the system. I just need to find out if at least one non-trivial solution exists, and if it turns out that one exists, or several, I'd like to be able to determine at least one of them, it doesn't matter which one, just as long as it is not the trivial solution, y_j = 0, for all j.

If one solution is known, can that solution be somehow 'removed' from the system, leaving a system sharing every other zero of the original system except the one which was removed? If so, is there an efficient algorithm to do this?

Is there a fast algorithm to find only 1-2 solutions for a system rather than every solution of the system?

I've looked around online a bit, and it seems like the problem of finding every solution of a system is quite computationally expensive. Also, I can only seem to find information concerning finding all solutions. Most stuff points towards computing a grobner basis or rational univarate representation for the system. That said, I can understand the computation of the grobner basis but it seems REALLY expensive. On the other hand I lack the education to understand how to compute the univariate representation, and do not want to take a long detour to learn it atm.

If anyone knows anything about the subject and could help me out I'd be very appreciative. Heck, even if you can just point me in the right direction I'd be happy.

I appreciate your time and consideration.

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# Remove Known Solution from System of Polynomials?

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