Polynomial Elimination

The fundamental problem in algebra is determine the values of $$ {X_1}, \ldots ,{X_n} $$ such that 9.1$$ \begin{array}{*{20}{c}} {{F_1}\left( {{X_1}, \ldots ,{X_n}} \right) = 0,} \\ \vdots \\ {{F_m}\left( {{X_1}, \ldots ,{X_n}} \right) = 0,} \end{array} $$ where the F _i are polynomials in the X _j . If the m = n = 1 then we have the familiar problem of finding the zeroes of single polynomial in one variable. One way to solve the (9.1) is to try use the constraints of one equation to eliminate variables from others. If this is done properly, we can (often) obtain a system of equations of the form 9.2$$ \begin{array}{*{20}{c}} {{G_1}\left( {{X_1}} \right) = 0,} \\ {{G_2}\left( {{X_1},{X_2}} \right) = 0,} \\ \vdots \\ {{G_m}\left( {{X_1}, \ldots ,{X_n}} \right) = 0.} \end{array}$$ By substituting each zero of the first equation into the others, and then solving the second and repeating we can obtain all of the solutions of (9.1).