Solution of Systems of Polynomial Equations

Given a system of equations of rational functions with exact coefficients

         p1(x1,...,xn)
             .
             .
         pm(x1,...,xn)

Axiom can find numeric or symbolic solutions. The system is first split into irreducible components, then for each component, a triangular system of equations is found that reduces the problem to sequential solutions of univariate polynomials resulting from substitution of partial solutions from the previous stage.

         q1(x1,...,xn)
             .
             .
         qm(xn)

Symbolic solutions can be presented using "implicit" algebraic numbers defined as roots of irreducible polynomials or in terms of radicals. Axiom can also find approximations to the real or complex roots of a system of polynomial equations to any user specified accuracy. The operation solve for systems is used in a way similar to solve for single equations. Instead of a polynomial equation, one has to give a list of equations and instead of a single variable to solve for, a list of variables. For solutions of single equations see Solution of a Single Polynomial Equation Use the operation solve if you want implicitly presented solutions.

Use radicalSolve if you want your solutions expressed in terms of radicals.

To get numeric solutions you only need to give the list of equations and the precision desired. The list of variables would be redundant information since there can be no parameters for the numerical solver. If the precision is expressed as a floating point number you get results expressed as floats.

To get complex numeric solutions, use the operation complexSolve, which takes the same arguments as in the real case.

It is also possible to solve systems of equations in rational functions over the rational numbers. Note that [x=0.0,a=0.0] is not returned as a solution since the denominator vanishes there.

When solving equations with denominators, all solutions where the denominator vanishes are discarded.