elimination of unknown

Consider the simultaneous polynomial equations

{a(x,y)=:i=0mai(y)xi= 0,b(x,y)=:j=0nbj(y)xj= 0 (1)

in two unknowns x and y, where e.g.  mn.  It is possible to eliminate one of the unknowns from (1), i.e. derive an equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/Equivalent3) pair of polynomial equations

{f(y)= 0,g(x,y)= 0.

First we form the polynomialPlanetmathPlanetmath

c(x,y)=:bn(y)a(x,y)-am(y)xm-nb(x,y), (2)

the degree of which is less than m.  When  (x0,y0)  is a solution of (1), then it satisfies

{c(x,y)= 0b(x,y)= 0. (3)

On the other hand, when  (x1,y1)  is a solution of (3), then (2) implies that it satisfies also (1), except possibly in the case  bn(y1)=0.
We can continue similarly until we arrive at a pair of equations

{f(y)= 0,g(x,y)= 0 (4)

Substituting the roots of the former of the equations (4) into the latter one, which in practice is usually of first degree with respect to x, one can get the corresponding values of x.  Hence one obtains all solutions of the original system of equations (1).  Since the cases  bn(y1)=0  may yield wrong solutions, one should check them by substituting into (1).

Note.  One can derive from the equations (1) an equation of lower degree also by eliminating from them the constant terms; the terms of resulting equation have as common factor x or its higher power, which is removed by dividing.

Title elimination of unknown
Canonical name EliminationOfUnknown
Date of creation 2013-03-22 19:20:27
Last modified on 2013-03-22 19:20:27
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type AlgorithmMathworldPlanetmath
Classification msc 26C05
Classification msc 13P10
Classification msc 12D99
Synonym elementary method of elimination
Related topic FactorizationOfPrimitivePolynomial