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Can I use resultant theory (or polynomial resultant method) to find solutions for four simultaneous polynomial equations with four unknown variables?

So far, I could only find examples which uses two equations having two unknown variables. I could also see an example of three unknown variables & three equations, but in that the first unknown was easily expressed as a function of other two variables.

The cases that I have come across for polynomials $f_1, f_2, ..., f_n $ are

A) Solve: $f_1(x_1, x_2)=0$ and $f_2(x_1, x_2)=0$

B) Solve: $f_1(x_1, x_2, x_3)=0$ and $f_2(x_1, x_2, x_3)=0$ and $x_1=g(x_2, x_3)$. Here, g(.) is known

I am looking for procedure/example for solving using resultant method cases like these:

C) Solve: $f_1(x_1, x_2, x_3)=0$ and $f_2(x_1, x_2, x_3)=0$ and $f_3(x_1, x_2, x_3)=0$

D) Solve: $f_1(x_1, x_2, x_3, x_4 )=0$ and $f_2(x_1, x_2, x_3, x_4)=0$ and $f_3(x_1, x_2, x_3, x_4)=0$ and $f_4(x_1, x_2, x_3, x_4)=0$

Thank you in advance for your kind help.

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    $\begingroup$ Please do not write in all caps. $\endgroup$ Oct 27, 2015 at 13:42
  • $\begingroup$ This sounds like homework, and you can solve this by the more general Grobner basis algorithm. $\endgroup$ Oct 27, 2015 at 14:48
  • $\begingroup$ Let's tackle C (and D can be done similarly). Take the resultants of $f_1,f_2$ and $f_1,f_3$ with respect to $x_3$. This gives you two polynomials in $x_1,x_2$. You can now take the resultant of these two polynomials with respect to $x_2$ and get a polynomial in $x_1$. The roots of this last polynomial correspond to the first coordinates of the solutions to your system and you can solve back for the other coordinates. In practice, this is not very efficient as the degrees grow and Grobner bases are better. $\endgroup$ Oct 27, 2015 at 16:08
  • $\begingroup$ Per Alexandersson, it is not homework. Thank you for pointing towards Grobner basis algorithm. I am actually looking for a software package or library which can solve a system of multivariate polynomial equations using resultant theory. If there isn't one, I would at least like to know if the general mathematical steps are known. I could only find resultant operation for 2 polynomial equations in 2 unknowns so far (in MuPAD and in Mathematica). I was wondering if there is a link or information about solving a larger system (say, 4 polynomials with 4 unknowns) using the resultant theory. $\endgroup$
    – Joy
    Oct 27, 2015 at 16:10
  • $\begingroup$ Felipe Voloch, thank you. I will try that (resultant of resultants). I was looking for some sort of pseudo-code to solve the system. You have already given me the direction. I will also try Grobner bases method, but might have to read more. The original intention was to optimize each polynomial such that they are simultaneously nearer to zero. Solving is the ideal case that might occur for a few sets of polynomials. Are there specific algorithms which can optimize a system of polynomials (so as to have near-zero values)? Thank you again. $\endgroup$
    – Joy
    Oct 27, 2015 at 16:16

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