Resultant (algebra)

	Main Article	Discussion	Related Articles  [?]	Bibliography  [?]	External Links  [?]	Citable Version  [?]
	This editable Main Article is under development and subject to a disclaimer. [edit intro]

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

Given polynomials

${\displaystyle f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0}$

and

${\displaystyle g(x)=b_m{x}^m+b_{m-1}{x}^{m-1}+\cdots +b_{1}x+b_0}$

with roots

${\displaystyle {\alpha }_1,\dots ,{\alpha }_n\text{ and }{\beta }_1,\dots ,{\beta }_m}$

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

${\displaystyle R(f,g)=a_n^m{b}_m^n{\displaystyle \prod _{i=1}^n}{\displaystyle \prod _{j=1}^m}({\alpha }_i-{\beta }_j).}$

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

Sylvester matrix[edit]

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

${\displaystyle \left(\begin{array}{ccccccc}{a}_n& a_{n-1}& \dots & a_0& 0& \dots & 0\\ 0& a_n& \dots & a_1& a_0& \dots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& \ddots & a_0\\ b_n& b_{n-1}& \dots & b_0& 0& \dots & 0\\ 0& b_n& \dots & b_1& b_0& \dots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& \ddots & b_0\end{array}\right)}$

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

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

and expanding the determinant we see that

${\displaystyle R(f,g)=a(X)f(X)+b(X)g(X)}$

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[edit]

• 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.