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I have a complicated set of multivariate polynomial equations, all of which are necessarily equal to zero. I had been using groebner bases as a method to solve these by finding the groebner basis (which is usually simpler than the original set) and then running Mathematica's "Solve" command on the resulting equations (setting each polynomial in the basis equal to zero). I am now wondering if this is really the best approach, since (as the number of equations and variables increase) groebner bases seem to be out of reach. Given that the equations are equal to zero, and that the groebner bases algorithm does not seem to be taking this into account, is there another algorithm more specifically designed for this situation?
When I say that the groebner bases algorithm doesn't take this into account, what I mean is the following:
I am expecting many of the variables will themselves be equal to zero (this is what has happened in previous computations, after solving the groebner bases). Let's say for a particular ideal over a ring 0,(x,y,z),dp that x is found to be in the groebner basis. From the algorithms that I see, this will only help eliminate the x variables which are the leading monomials in the remaining polynomials under consideration. Ideally, if the algorithm took into account that this means that x=0, it could greatly simplify the remaining polynomials. It would also be beneficial if the algorithm would also take into account situations where x-y is found in the basis (setting x=y everywhere) or x-c is in the basis, where c is a constant.
Sometimes even though x is not found in the groebner basis, it is found out that x=0 after solving. To my thinking, a better approach than the groebner basis algorithm for this case would be one which focused on converting multivariate polynomials into univariate polynomials. This probably does not have appeal in general, but I think in this particular case it would be useful.
Now that I have rambled far off course, I should summarize what is really my question:
Given a set of multivariate polynomial equations equal to zero, what are the methods available to find (all or some) real roots of the system?
I have a complicated set of multivariate polynomial equations, all of which are necessarily equal to zero. I had been using groebner bases as a method to solve these by finding the groebner basis (which is usually simpler than the original set) and then running Mathematica's "Solve" command on the resulting equations (setting each polynomial in the basis equal to zero). I am now wondering if this is really the best approach, since (as the number of equations and variables increase) groebner bases seem to be out of reach. Given that the equations are equal to zero, and that the groebner bases algorithm does not seem to be taking this into account,[b] is there another algorithm more specifically designed for this situation?[/b]
When I say that the groebner bases algorithm doesn't take this into account, what I mean is the following:
I am expecting many of the variables will themselves be equal to zero (this is what has happened in previous computations, after solving the groebner bases). Let's say for a particular ideal over a ring 0,(x,y,z),dp that x is found to be in the groebner basis. From the algorithms that I see, this will only help eliminate the x variables which are the leading monomials in the remaining polynomials under consideration. Ideally, if the algorithm took into account that this means that x=0, it could greatly simplify the remaining polynomials. It would also be beneficial if the algorithm would also take into account situations where x-y is found in the basis (setting x=y everywhere) or x-c is in the basis, where c is a constant.
Sometimes even though x is not found in the groebner basis, it is found out that x=0 after solving. To my thinking, a better approach than the groebner basis algorithm for this case would be one which focused on converting multivariate polynomials into univariate polynomials. This probably does not have appeal in general, but I think in this particular case it would be useful.
Now that I have rambled far off course, I should summarize what is really my question:
[b]Given a set of multivariate polynomial equations equal to zero, what are the methods available to find (all or some) [i]real[/i] roots of the system?[/b]
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