Resultant (algebra)

In algebra, the resultant of two polynomials is a quantity which determines whether or not they have a factor in common.

Given polynomials

$f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}$

and

$g(x)=b_{m}x^{m}+b_{m-1}x^{m-1}+\cdots +b_{1}x+b_{0}$

with roots

$\alpha _{1},\ldots ,\alpha _{n}{\mbox{ and }}\beta _{1},\ldots ,\beta _{m}$

respectively, the resultant R(f,g) with respect to the variable x is defined as

$R(f,g)=a_{n}^{m}b_{m}^{n}\prod _{i=1}^{n}\prod _{j=1}^{m}(\alpha _{i}-\beta _{j}).$

The resultant is thus zero if and only if f and g have a common root.

Sylvester matrix

The Sylvester matrix attached to f and g is the square (m+n)×(m+n) matrix

${\begin{pmatrix}a_{n}&a_{n-1}&\ldots &a_{0}&0&\ldots &0\\0&a_{n}&\ldots &a_{1}&a_{0}&\ldots &0\\\vdots &\vdots &\ddots &\vdots &\vdots &\ddots &a_{0}\\b_{n}&b_{n-1}&\ldots &b_{0}&0&\ldots &0\\0&b_{n}&\ldots &b_{1}&b_{0}&\ldots &0\\\vdots &\vdots &\ddots &\vdots &\vdots &\ddots &b_{0}\end{pmatrix}}$

in which the coefficients of f occupy m rows and those of g occupy n rows.

The determinant of the Sylvester matrix is the resultant of f and g.

References

• J.W.S. Cassels (1991). Lectures on Elliptic Curves. Cambridge University Press. ISBN 0-521-42530-1.  Chapter 16.