On Friday, May 2, 2014 7:28:30 PM UTC-4, roger.d....@gmail.com wrote: > Hello, are there standard methods for solving systems of multi-variable equations where all variables are of order one, such as the following?: > > > > c11.xyz + c12.xy + c13.yz + c14.xz + c15.x + c16.y + c17.z = c10 > > c21.xyz + c22.xy + c23.yz + c24.xz + c25.x + c26.y + c27.z = c30 > > c31.xyz + c32.xy + c33.yz + c34.xz + c35.x + c36.y + c37.z = c30 > > > > There are always an equal number of equations and variables, which in my application may be dozens of variables. Of possible importance, a matrix (or multiple matrices of multiple dimensions) of the constants would be sparse. Any references to standard solutions would be greatly appreciated. These might include numerical approaches. Thanks in advance for any help! > >
Usually the best way to solve systems of polynomial equations symbolically is with resultants, especially the Dixon resultant. It is usually much much better than trying Groebner bases. The resultant is a single polynomial in one of the variables; the others are eliminated. Above, you would eliminate, say, y and z, leaving a polynomial in x.
If you want a fully symbolic solution of the system above, with all 24 parameters, I think that would be doable.
But if you want many more equations, I think a fully symbolic solution is unfeasible. If you mean that most of the parameters will be 0, then there is a good chance for a fully symbolic solution for maybe up to 10 equations.