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.

The rows of the Sylvester matrix may be interpreted as the coefficients of the polynomials

$X^{0}f,X^{1}f,\ldots ,X^{m-1}f,X^{0}g,X^{1}g,\ldots ,X^{n-1}g\,$

and expanding the determinant we see that

$R(f,g)=a(X)f(X)+b(X)g(X)$

with a and b polynomials of degree at most m-1 and n-1 respectively, and R a scalar. If f and g have a polynomial common factor this must divide R and so R must be zero. Conversely if R is zero, then f/g = - b/a so f/g is not in lowest terms and f and g have a common factor.

References

• J.W.S. Cassels (1991). Lectures on Elliptic Curves. Cambridge University Press. ISBN 0-521-42530-1.  Chapter 16.
• Serge Lang (1993). Algebra, 3rd ed. Addison-Wesley, 200-204. ISBN 0-201-55540-9.