Resultant (algebra)

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This editable Main Article is under development and subject to a disclaimer.

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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 + ⋯ + a 1 x + a 0 {\displaystyle 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 + ⋯ + b 1 x + b 0 {\displaystyle g(x)=b_{m}x^{m}+b_{m-1}x^{m-1}+\cdots +b_{1}x+b_{0}}

with roots

α 1 , … , α n and β 1 , … , β m {\displaystyle \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 ∏ i = 1 n ∏ j = 1 m ( α i − β j ) . {\displaystyle 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[edit]

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

( a n a n − 1 … a 0 0 … 0 0 a n … a 1 a 0 … 0 ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ a 0 b n b n − 1 … b 0 0 … 0 0 b n … b 1 b 0 … 0 ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ b 0 ) {\displaystyle {\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 , … , X m − 1 f , X 0 g , X 1 g , … , X n − 1 g {\displaystyle 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 ) {\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.