---
tag: 8D277VG29T
kind: definition
name: Resultant
msc: [13P15, "12E05"]
defines: NNPHTWF645
uses: [0GYGHKX834, 0Q9EZS90GY, 0C3CFHDN6R, 02NZZ1J0FH]
sources:
  - url: https://en.wikipedia.org/wiki/Resultant
---

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]], and let $m, n \ge 0$ be integers with $m + n \ge 1$.
Let $f = a_m x^m + \dots + a_0$ and $g = b_n x^n + \dots + b_0$ be elements of $R[x]$.
The Sylvester matrix $S(f, g)$ is the $(m + n) \times (m + n)$ matrix whose rows are:

1. for $i = 1, \dots, n$, the row $(a_m, a_{m-1}, \dots, a_0)$ preceded by $i - 1$ zeros and filled up with zeros on the right;
2. for $j = 1, \dots, m$, the row $(b_n, b_{n-1}, \dots, b_0)$ preceded by $j - 1$ zeros and filled up with zeros on the right.

The resultant of $f$ and $g$ is $\operatorname{Res}(f, g) = \det S(f, g)$, the [[determinant|0C3CFHDN6R]] of this matrix.

## Remarks

Example: for $f = x - a$ and $g = x - b$, the matrix is $\begin{pmatrix} 1 & -a \\ 1 & -b \end{pmatrix}$, and $\operatorname{Res}(f, g) = a - b$.
Let $R = K$ be a field with $a_m \ne 0$ and $b_n \ne 0$, and let $\alpha_1, \dots, \alpha_m$ and $\beta_1, \dots, \beta_n$ be the roots of $f$ and $g$ in a [[splitting field|02NZZ1J0FH]] of $f g$.
Then
$$\operatorname{Res}(f, g) = a_m^n \, b_n^m \prod_{i=1}^{m} \prod_{j=1}^{n} (\alpha_i - \beta_j).$$
So $\operatorname{Res}(f, g) = 0$ if and only if $f$ and $g$ have a common root, if and only if they have a common factor of positive degree.
For instance, $\operatorname{Res}(x^2 - 1, x - 1) = 0$, because of the common root $1$.
The value depends on the chosen formal degrees $m$ and $n$: if $a_m = b_n = 0$, the first column is zero and the resultant is $0$ even without a common root.
For $f$ of degree $n$ with leading coefficient $a_n$, $\operatorname{Res}(f, f') = (-1)^{n(n-1)/2} a_n \operatorname{Disc}(f)$, where $\operatorname{Disc}$ is the [[discriminant|0CCPN09EMJ]].

## Explanations

### L1 human

The resultant is the determinant of a square table filled with shifted copies of the two coefficient lists.
It is zero exactly when the two polynomials share a root, as long as both leading coefficients are nonzero.