example of resultant (2)
This example shows how resultants can be used to solve simultaneous algebraic equations in two variables. We shall compute the intersection of two ellipses.
Consider the system of equations where
We will consider and as polynomials in whose coefficients are functions of . What this means can be seen by writing and as
We will now construct the resultant by computing Sylvester’s determinant. In the notation of the main article, the coefficients of the various powers of may be notated as
The determinant is
This determinant evaluates to . Hence, in order for the system of equations to have a solution, must satisfy the equation
We can factor the polynomial as
Hence, the solutions are
Note that the solution occurs with multiplicity . We shall see what that means shortly.
Having found the possible values of , let us now find the corresponding values for . Substituting the possible value into the equation , we obtain
Hence, either or . If we substitute and into , we obtain zero so
is a solution of our system. However, if we substitute and into , we obtain , so this root does not lead to a solution of the original system of equations.
Substituting the possible value into the equation , we obtain
Hence, either or . If we substitute and into , we obtain zero so
is a solution of our system. However, if we substitute and into , we obtain , so this root does not lead to a solution of the original system of equations.
Finally, let us consider the value . Substituting this value into , we obtain the equation
This equation has the solutions and . Substituting and into , we obtain , so
is a solution of the system of equations. Likewise, substituting and into , we obtain , so
is a solution of the system of equations. In this case, we obtained two solutions to the system of equations.
At this point, recall the remark that was a double root of the resultant. This fact explains why both values of gave rise to solutions of the system when . In general, the number of solutions (counted with multiplicity) of the system of equations for a particular value of equals the mutiplicity of that value of as a root of the resultant.
Title | example of resultant (2) |
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Canonical name | ExampleOfResultant2 |
Date of creation | 2013-03-22 14:36:36 |
Last modified on | 2013-03-22 14:36:36 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 7 |
Author | rspuzio (6075) |
Entry type | Example |
Classification | msc 13P10 |