In
mathematics, the resultant of two
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
s is a
polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common
root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
(possibly in a
field extension), or, equivalently, a common factor (over their field of coefficients). In some older texts, the resultant is also called the eliminant.
The resultant is widely used in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
, either directly or through the
discriminant, which is essentially the resultant of a polynomial and its derivative. The resultant of two polynomials with
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
or polynomial coefficients may be computed efficiently on a computer. It is a basic tool of
computer algebra, and is a built-in function of most
computer algebra system
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The d ...
s. It is used, among others, for
cylindrical algebraic decomposition
In mathematics, cylindrical algebraic decomposition (CAD) is a notion, and an algorithm to compute it, that are fundamental for computer algebra and real algebraic geometry. Given a set ''S'' of polynomials in R''n'', a cylindrical algebraic decom ...
,
integration of
rational functions and drawing of
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
s defined by a
bivariate polynomial equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
.
The resultant of ''n''
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
s in ''n'' variables (also called multivariate resultant, or Macaulay's resultant for distinguishing it from the usual resultant) is a generalization, introduced by
Macaulay, of the usual resultant. It is, with
Gröbner bases, one of the main tools of
elimination theory
Elimination may refer to:
Science and medicine
*Elimination reaction, an organic reaction in which two functional groups split to form an organic product
*Bodily waste elimination, discharging feces, urine, or foreign substances from the body ...
.
Notation
The resultant of two univariate polynomials and is commonly denoted
or
In many applications of the resultant, the polynomials depend on several indeterminates and may be considered as univariate polynomials in one of their indeterminates, with polynomials in the other indeterminates as coefficients. In this case, the indeterminate that is selected for defining and computing the resultant is indicated as a subscript:
or
The degrees of the polynomials are used in the definition of the resultant. However, a polynomial of degree may also be considered as a polynomial of higher degree where the leading coefficients are zero. If such a higher degree is used for the resultant, it is usually indicated as a subscript or a superscript, such as
or
Definition
The resultant of two
univariate polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
s over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
or over a
commutative ring is commonly defined as the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of their
Sylvester matrix In mathematics, a Sylvester matrix is a matrix associated to two univariate polynomials with coefficients in a field or a commutative ring. The entries of the Sylvester matrix of two polynomials are coefficients of the polynomials. The determinant ...
. More precisely, let
:
and
:
be nonzero polynomials of degrees and respectively. Let us denote by
the
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
(or
free module if the coefficients belong to a commutative ring) of dimension whose elements are the polynomials of degree strictly less than . The map
:
such that
:
is a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
between two spaces of the same dimension. Over the basis of the powers of (listed in descending order), this map is represented by a square matrix of dimension , which is called the ''Sylvester matrix'' of and (for many authors and in the article
Sylvester matrix In mathematics, a Sylvester matrix is a matrix associated to two univariate polynomials with coefficients in a field or a commutative ring. The entries of the Sylvester matrix of two polynomials are coefficients of the polynomials. The determinant ...
, the Sylvester matrix is defined as the transpose of this matrix; this convention is not used here, as it breaks the usual convention for writing the matrix of a linear map).
The resultant of and is thus the determinant
:
which has columns of and columns of (the fact that the first column of 's and the first column of 's have the same length, that is , is here only for simplifying the display of the determinant).
For instance, taking and we get
:
If the coefficients of the polynomials belong to an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
, then
:
where
and
are respectively the roots, counted with their multiplicities, of and in any
algebraically closed field containing the integral domain.
This is a straightforward consequence of the characterizing properties of the resultant that appear below. In the common case of integer coefficients, the algebraically closed field is generally chosen as the field of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s.
Properties
In this section and its subsections, and are two polynomials in of respective degrees and , and their resultant is denoted
Characterizing properties
The following properties hold for the resultant of two polynomials with coefficients in
a
commutative ring . If is a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
or more generally an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
, the resultant is the unique function of the coefficients of two polynomials that satisfies these properties.
* If is a
subring of another ring , then
That is and have the same resultant when considered as polynomials over or .
*If (that is if
is a nonzero constant) then
Similarly, if , then
*
*
*
Zeros
* The resultant of two polynomials with coefficients in an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
is zero if and only if they have a
common divisor of positive degree.
* The resultant of two polynomials with coefficients in an integral domain is zero if and only if they have a common root in an
algebraically closed field containing the coefficients.
* There exists a polynomial of degree less than and a polynomial of degree less than such that
This is a generalization of
Bézout's identity
In mathematics, Bézout's identity (also called Bézout's lemma), named after Étienne Bézout, is the following theorem:
Here the greatest common divisor of and is taken to be . The integers and are called Bézout coefficients for ; they ...
to polynomials over an arbitrary commutative ring. In other words, the resultant of two polynomials belongs to the
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
generated by these polynomials.
Invariance by ring homomorphisms
Let and be two polynomials of respective degrees and with coefficients in a
commutative ring , and
a
ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is:
:addition preser ...
of into another commutative ring . Applying
to the coefficients of a polynomial extends
to a homomorphism of polynomial rings