Resultant (algebra)

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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

f(x)=anxn+an−1xn−1+⋯+a1x+a0

and

g(x)=bmxm+bm−1xm−1+⋯+b1x+b0

with roots

α1,…,αn and β1,…,βm

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

R(f,g)=anmbmn∏i=1n∏j=1m(αi−βj).

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

(anan−1…a00…00an…a1a0…0⋮⋮⋱⋮⋮⋱a0bnbn−1…b00…00bn…b1b0…0⋮⋮⋱⋮⋮⋱b0)

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

X0f,X1f,…,Xm−1f,X0g,X1g,…,Xn−1g

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[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.