Contents - Student subdomain for University of Bath

Appendix A
Algebraic Background
A.1 The resultant
It quite often happens that we have to consider whether two polynomials, which
are usually relatively prime, can have a common factor in certain special cases.
The basic algebraic tool for solving this problem is called the resultant. In this
section we shall define this object and we shall give some properties of it. We
take the case of two polynomials f and g in one variable x and with coefficients
in a ring R.
We write f = ∑ n
i=0 a ix i and g = ∑ m
i=0 b ix i .
Definition 88 The Sylvester matrix of f and g is the matrix
⎛
a n a n−1 . . . a 1 a 0 0 0 . . . 0
⎞
0 a n a n−1 . . . a 1 a 0 0 . . . 0
. . .. . .. . .. . . . . .. . .. . .. .
0 . . . 0 a n a n−1 . . . a 1 a 0 0
0 . . . 0 0 a
Syl(f, g) =
n a n−1 . . . a 1 a 0
b m b m−1 . . . b 1 b 0 0 0 . . . 0
0 b m b m−1 . . . b 1 b 0 0 . . . 0
.
⎜ . .. . .. . .. . . . . .. . .. . .. . ⎟
⎝
⎠
0 . . . 0 b m b m−1 . . . b 1 b 0 0
0 . . . 0 0 b m b m−1 . . . b 1 b 0
where there are m lines constructed with the a i , n lines constructed with the b i .
Definition 89 The resultant of f and g, written Res(f, g), or Res x (f, g) if
there is doubt about the variable, is the determinant of this matrix.
Well-known properties of determinants imply that the resultant belongs to R,
and that Res(f, g) and Res(g, f) are equal, to within a sign. We must note that,
although the resultant is defined by a determinant, this is not the best way
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