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{{Short description|Mathematical concept in polynomial theory}}
{{about|the resultant of two polynomials|the uses as an adjective|Resultant (disambiguation)}}

In [[mathematics]], the '''resultant''' of two [[polynomial]]s is a [[polynomial expression]] of their coefficients that is equal to zero if and only if the polynomials have a common [[root of a function|root]] (possibly in a [[field extension]]), or, equivalently, a common [[Factorization|factor]] (over their field of coefficients). In some older texts, the resultant is also called the '''eliminant'''.<ref>{{Citation | last1=Salmon | first1=George | title=Lessons introductory to the modern higher algebra | orig-year=1859 | url=https://archive.org/details/salmonalgebra00salmrich | publisher=Dublin, Hodges, Figgis, and Co. | edition=4th | isbn=978-0-8284-0150-0 | year=1885 | at = lesson VIII, p. 66 }}</ref>

The resultant is widely used in [[number theory]], either directly or through the [[discriminant]], which is essentially the resultant of a polynomial and its [[derivative]]. The resultant of two polynomials with [[rational number|rational]] 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]]s. It is used, among others, for [[cylindrical algebraic decomposition]], [[Symbolic integration|integration]] of [[rational function]]s and drawing of [[curve]]s defined by a [[Polynomial#Number of variables|bivariate]] [[polynomial equation]].

The resultant of ''n'' [[homogeneous polynomial]]s in ''n'' variables (also called '''multivariate resultant''', or '''Macaulay's resultant''' for distinguishing it from the usual resultant) is a generalization, introduced by [[Francis Sowerby Macaulay|Macaulay]], of the usual resultant.<ref>{{citation|first=F. S.|last= Macaulay|author-link=Francis Sowerby Macaulay|title=Some Formulæ in Elimination|journal=[[Proc. London Math. Soc.]] | year=1902 | volume=35 | pages= 3–27|doi=10.1112/plms/s1-35.1.3 |url= https://zenodo.org/record/1447750}}</ref> It is, with [[Gröbner basis|Gröbner bases]], one of the main tools of [[elimination theory]].

==Notation==
The resultant of two univariate polynomials {{math|''A''}} and {{math|''B''}} is commonly denoted <math>\operatorname{res}(A,B)</math> or <math>\operatorname{Res}(A,B).</math>

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: <math>\operatorname{res}_x(A,B)</math> or <math>\operatorname{Res}_x(A,B).</math>

The [[Degree of a polynomial|degree]]s of the polynomials are used in the definition of the resultant. However, a polynomial of degree {{math|''d''}} 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 <math>\operatorname{res}_{d,e}(A,B)</math> or <math>\operatorname{res}_x^{d,e}(A,B).</math>

==Definition==

The '''resultant''' of two [[univariate polynomial]]s over a [[field (mathematics)|field]] or over a [[commutative ring]] is commonly defined as the [[determinant]] of their [[Sylvester matrix]]. More precisely, let 
<math display="block">A = a_0 x^d + a_1 x^{d-1} + \cdots + a_d</math>
and
<math display="block">B = b_0 x^e + b_1 x^{e-1} + \cdots + b_e</math>
be nonzero polynomials of degrees {{math|''d''}} and {{math|''e''}} respectively. Let us denote by <math>\mathcal{P}_i</math> the [[vector space]] (or [[free module]] if the coefficients belong to a commutative ring) of dimension {{math|''i''}} whose elements are the polynomials of degree strictly less than {{math|''i''}}. The map
<math display="block">\varphi:\mathcal{P}_{e}\times \mathcal{P}_{d} \rightarrow \mathcal{P}_{d+e}</math> 
such that
<math display="block">\varphi(P,Q) = AP + BQ</math>
is a [[linear map]] between two spaces of the same dimension. Over the [[vector space basis|basis]] of the powers of {{math|''x''}} (listed in descending order), this map is represented by a square matrix of dimension {{math|''d'' + ''e''}}, which is called the ''Sylvester matrix'' of {{math|''A''}} and {{math|''B''}} (for many authors and in the article [[Sylvester matrix]], 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 {{math|''A''}} and {{math|''B''}} is thus the determinant
<math display="block">\begin{vmatrix} 
a_0      & 0           & \cdots & 0          & b_0        & 0              & \cdots & 0       \\
a_1    & a_0       & \cdots & 0           & b_1     & b_0           & \cdots & 0  \\
a_2    & a_1     & \ddots & 0           & b_2     & b_1         & \ddots & 0 \\
\vdots  &\vdots   & \ddots & a_0        & \vdots   &\vdots       & \ddots & b_0  \\
a_d       & a_{d-1} & \cdots & \vdots   & b_e       & b_{e-1}     & \cdots & \vdots\\
0          & a_d       & \ddots &  \vdots  & 0          & b_e          & \ddots &  \vdots  \\
\vdots  & \vdots   & \ddots & a_{d-1}  & \vdots  & \vdots      & \ddots & b_{e-1}   \\
0          & 0          & \cdots  & a_d       & 0           & 0              & \cdots & b_e   
\end{vmatrix},</math>
which has {{math|''e''}} columns of {{math|''a''<sub>''i''</sub>}} and {{math|''d''}} columns of {{math|''b''<sub>''j''</sub>}} (the fact that the first column of {{mvar|a}}'s and the first column of {{mvar|b}}'s have the same length, that is {{math|1=''d'' = ''e''}}, is here only for simplifying the display of the determinant).
For instance, taking {{math|1=''d'' = 3}} and {{math|1=''e'' = 2}} we get
<math display="block">\begin{vmatrix} 
a_0    & 0     & b_0   & 0    & 0 \\
a_1    & a_0   & b_1   & b_0  & 0  \\
a_2    & a_1   & b_2   & b_1  & b_0 \\
a_3    & a_2   & 0     & b_2  & b_1 \\
0      & a_3   & 0     & 0    & b_2
\end{vmatrix}.</math>

If the coefficients of the polynomials belong to an [[integral domain]], then 
<math display="block">\operatorname{res}(A, B) = a_0^e b_0^d \prod_{\begin{array}{c}1 \leq i \leq d\\ 1 \leq j \leq e\end{array}} (\lambda_i-\mu_j) = a_0^e \prod_{i=1}^d B(\lambda_i) = (-1)^{de} b_0^d \prod_{j=1}^e A(\mu_j),</math>
where <math>\lambda_1, \dots, \lambda_d</math> and <math>\mu_1,\dots,\mu_e</math> are respectively the roots, counted with their multiplicities, of {{mvar|A}} and {{mvar|B}} 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]]s.

==Properties==
In this section and its subsections, {{math|''A''}} and {{math|''B''}} are two polynomials in {{math|''x''}} of respective degrees {{math|''d''}} and {{math|''e''}}, and their resultant is denoted 
<math>\operatorname{res}(A,B).</math>

===Characterizing properties===
The following properties hold for the resultant of two polynomials with coefficients in 
a [[commutative ring]] {{math|''R''}}. If {{mvar|R}} is a [[field (mathematics)|field]] or more generally an [[integral domain]], the resultant is the unique function of the coefficients of two polynomials that satisfies these properties.

* If {{mvar|R}} is a [[subring]] of another ring {{mvar|S}}, then <math>\operatorname{res}_R(A,B) = \operatorname{res}_S(A,B).</math> That is {{mvar|A}} and {{mvar|B}} have the same resultant when considered as polynomials over {{mvar|R}} or {{mvar|S}}.
*If {{math|1=''d'' = 0}} (that is if <math>A=a_0</math> is a nonzero constant) then <math>\operatorname{res}(A,B) = a_0^e.</math> Similarly, if {{math|1=''e'' = 0}}, then <math>\operatorname{res}(A,B) = b_0^d.</math>
* <math>\operatorname{res}(x+a_1, x+b_1) = b_1-a_1</math>
* <math>\operatorname{res}(B,A)=(-1)^{de} \operatorname{res}(A,B)</math> 
* <math>\operatorname{res}(AB,C) = \operatorname{res}(A,C)\operatorname{res}(B,C)</math>

===Zeros===
* The resultant of two polynomials with coefficients in an [[integral domain]] {{math|''D''}} is zero if and only if they have a [[Greatest common divisor of two polynomials|common divisor]] of positive degree over the [[field of fractions]] of {{math|''D''}}.{{efn|Note that if {{math|''D''}} is not a UFD, there may be polynomials <math>A</math>, <math>B</math> with <math>\operatorname{res}(A,B)=0</math>, but with no common factors of positive degree in {{math|''D''}}. One example in <math>\mathbf{Z}\left[\sqrt{-5}\right]</math> is <math>A=2x-(1+i\sqrt{5})</math>, <math>B= (1+i\sqrt{5})x-(-2+i\sqrt{5})</math>, with a common root of <math>(1+i\sqrt{5})/2 </math> but without common factors {{!=}} 1,-1.}}
* 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 {{math|''P''}} of degree less than {{math|''e''}} and a polynomial {{math|''Q''}} of degree less than {{math|''d''}} such that <math> \operatorname{res}(A,B)=AP+BQ.</math> This is a generalization of [[Bézout's identity]] to polynomials over an arbitrary commutative ring.  In other words, the resultant of two polynomials belongs to the [[ideal (ring theory)|ideal]] generated by these polynomials.

===Invariance by ring homomorphisms===
Let {{math|''A''}} and {{math|''B''}} be two polynomials of respective degrees {{math|''d''}} and {{math|''e''}} with coefficients in a [[commutative ring]] {{math|''R''}}, and <math>\varphi\colon R\to S</math> a [[ring homomorphism]] of {{math|''R''}} into another commutative ring {{math|''S''}}. Applying <math>\varphi</math> to the coefficients of a polynomial extends <math>\varphi</math> to a homomorphism of polynomial rings <math>R[x]\to S[x]</math>, which is also denoted <math>\varphi.</math> With this notation, we have:
* If <math>\varphi</math> preserves the degrees of  {{math|''A''}} and {{math|''B''}} (that is if <math>\deg(\varphi(A)) = d</math> and <math>\deg(\varphi(B)) = e</math>), then <math display="block">\varphi(\operatorname{res}(A,B)) = \operatorname{res}(\varphi(A), \varphi(B)).</math>
* If <math>\deg(\varphi(A)) < d</math> and <math>\deg(\varphi(B))< e,</math> then <math display="block">\varphi(\operatorname{res}(A,B)) = 0.</math>
* If <math>\deg(\varphi(A)) = d</math> and <math>\deg(\varphi(B)) =f < e,</math>  and the leading coefficient of {{math|''A''}} is <math>a_0</math> then <math display="block">\varphi(\operatorname{res}(A,B))=\varphi(a_0)^{e-f}\operatorname{res}(\varphi(A), \varphi(B)).</math>
* If <math>\deg(\varphi(A)) = f<d</math> and <math>\deg(\varphi(B)) = e,</math>  and the leading coefficient of {{math|''B''}} is <math>b_0</math> then <math display="block">\varphi(\operatorname{res}(A,B)) = (-1)^{e(d-f)}\varphi(b_0)^{d-f}\operatorname{res}(\varphi(A), \varphi(B)).</math>

These properties are easily deduced from the definition of the resultant as a determinant. They are mainly used in two situations. For computing a resultant of polynomials with integer coefficients, it is generally faster to compute it [[modular arithmetic|modulo]] several primes and to retrieve the desired resultant with [[Chinese remainder theorem]]. When {{math|''R''}} is a polynomial ring in other indeterminates, and {{math|''S''}} is the ring obtained by specializing to numerical values some or all indeterminates of {{math|''R''}}, these properties may be restated as ''if the degrees are preserved by the specialization, the resultant of the specialization of two polynomials is the specialization of the resultant''. This property is fundamental, for example, for [[cylindrical algebraic decomposition]].

===Invariance under change of variable===
*<math>\operatorname{res}(A(x+a), B(x+a)) = \operatorname{res}(A(x), B(x))</math>
*<math>\operatorname{res}(A(ax), B(ax)) = a^{de}\operatorname{res}(A(x), B(x))</math>
* If <math>A_r(x) = x^d A(1/x)</math> and <math>B_r(x) = x^e B(1/x)</math> are the [[reciprocal polynomial]]s of {{math|''A''}} and {{math|''B''}}, respectively, then <math display="block">\operatorname{res}(A_r, B_r) = (-1)^{de} \operatorname{res}(A,B)</math>

This means that the property of the resultant being zero is invariant under linear and projective changes of the variable.

===Invariance under change of polynomials===
*If {{math|''a''}} and {{mvar|''b''}} are nonzero constants (that is they are independent of the indeterminate {{math|''x''}}), and {{math|''A''}} and {{mvar|''B''}} are as above, then <math display="block">\operatorname{res}(aA,bB) =a^eb^d\operatorname{res}(A,B). </math>
*If {{math|''A''}} and {{mvar|''B''}} are as above, and {{mvar|C}} is another polynomial such that the degree of {{math|''A'' – ''CB''}} is {{math|''{{delta}}''}}, then  <math display="block">\operatorname{res}(B, A-CB)=b_0^{\delta-d}\operatorname{res}(B,A). </math> 

It is only when {{tmath|BC}} and {{tmath|A}} have the same degree that {{tmath|\delta}} cannot be deduced from the degrees of the given polynomials. If either {{mvar|B}} is [[monic polynomial|monic]], or {{math|deg ''C'' < deg ''A'' – deg ''B''}}, then <math display="block">\operatorname{res}(B,A-CB) = \operatorname{res}(B,A), </math> If {{math|1=''f'' = deg ''C'' > deg ''A'' – deg ''B'' = ''d'' – ''e''}}, then <math display="block">\operatorname{res}(B,A-CB)=b_0^{e+f-d}\operatorname{res}(B,A). </math>

These properties imply that in the [[Euclidean algorithm for polynomials]], and all its variants ([[pseudo-remainder sequence]]s), the resultant of two successive remainders (or pseudo-remainders) differs from the resultant of the initial polynomials by a factor which is easy to compute. Conversely, this allows one to deduce the resultant of the initial polynomials from the value of the last remainder or pseudo-remainder. This is the starting idea of the [[Polynomial greatest common divisor#Subresultant sequence with pseudo-remainders|subresultant-pseudo-remainder-sequence algorithm]], which uses the above formulae for getting [[subresultant|subresultant polynomials]] as pseudo-remainders, and the resultant as the last nonzero pseudo-remainder (provided that the resultant is not zero). This algorithm works for polynomials over the integers or, more generally, over an integral domain, without any division other than exact divisions (that is, without involving fractions). It involves <math>O(de)</math> arithmetic operations, while the computation of the determinant of the Sylvester matrix with standard algorithms requires <math>O((d+e)^3)</math> arithmetic operations.

===Generic properties===
In this section, we consider two polynomials
<math display="block">A = a_0x^d +a_1x^{d-1} + \cdots + a_d</math>
and
<math display="block">B = b_0x^e +b_1x^{e-1} + \cdots + b_e</math>
whose {{math|''d'' + ''e'' + 2}} coefficients are distinct [[indeterminate (variable)|indeterminate]]s. Let
<math display="block">R = \mathbb{Z}[a_0, \ldots, a_d, b_0, \ldots, b_e]</math>
be the polynomial ring over the integers defined by these indeterminates.
The resultant <math>\operatorname{res}(A,B)</math> is often called the '''generic resultant''' for the degrees {{math|''d''}} and {{math|''e''}}. It has the following properties.

*<math>\operatorname{res}(A,B)</math> is an [[absolutely irreducible]] polynomial.
*If <math>I</math> is the [[ideal (ring theory)|ideal]] of <math>R[x]</math> generated by {{math|''A''}} and {{math|''B''}}, then <math>I\cap R</math> is the [[principal ideal]] generated by <math>\operatorname{res}(A,B)</math>.

===Homogeneity===
The generic resultant for the degrees {{math|''d''}} and {{math|''e''}} is [[homogeneous polynomial|homogeneous]] in various ways. More precisely:
* It is homogeneous of degree {{math|''e''}} in <math>a_0, \ldots, a_d.</math>
* It is homogeneous of degree {{math|''d''}} in <math>b_0, \ldots, b_e.</math>
* It is homogeneous of degree {{math|''d'' + ''e''}} in all the variables <math>a_i</math> and <math>b_j.</math>
* If <math>a_i</math> and <math>b_i</math> are given the weight {{math|''i''}} (that is, the weight of each coefficient is its degree as [[elementary symmetric polynomial]]), then it is [[quasi-homogeneous polynomial|quasi-homogeneous]] of total weight {{math|''de''}}.
*If {{math|''P''}} and {{math|''Q''}} are homogeneous multivariate polynomials of respective degrees {{math|''d''}} and {{math|''e''}}, then their resultant in degrees {{math|''d''}} and {{math|''e''}} with respect to an indeterminate {{math|''x''}}, denoted <math>\operatorname{res}_x^{d,e}(P,Q)</math> in {{slink||Notation}}, is homogeneous of degree {{math|''de''}} in the other indeterminates.

===Elimination property===
Let <math>I=\langle A, B\rangle </math> be the [[ideal (ring theory)|ideal]] generated by two polynomials {{math|''A''}} and {{math|''B''}} in a polynomial ring <math>R[x],</math> where <math>R=k[y_1,\ldots,y_n]</math> is itself a polynomial ring over a field. If at least one of {{math|''A''}} and {{math|''B''}} is [[monic polynomial|monic]] in {{mvar|x}}, then:
* <math>\operatorname{res}_x(A,B)\in I \cap R</math>
* The ideals <math>I\cap R</math> and <math>R\operatorname{res}_x(A,B)</math> define the same [[algebraic set]]. That is, a tuple of {{math|''n''}} elements of an [[algebraically closed field]] is a common zero of the elements of <math>I\cap R</math> if and only it is a zero of <math>\operatorname{res}_x(A,B).</math> 
* The ideal <math>I\cap R</math> has the same [[radical of an ideal|radical]] as the [[principal ideal]] <math>R\operatorname{res}_x(A,B).</math> That is, each element of <math>I\cap R</math> has a power that is a multiple of <math>\operatorname{res}_x(A,B).</math>
* All [[irreducible polynomial|irreducible factor]]s of <math>\operatorname{res}_x(A,B)</math> divide every element of <math>I\cap R.</math>

The first assertion is a basic property of the resultant. The other assertions are immediate corollaries of the second one, which can be proved as follows.

As at least one of {{math|''A''}} and {{math|''B''}} is monic, a tuple <math>(\beta_1,\ldots, \beta_n)</math> is a zero of <math>\operatorname{res}_x(A,B)</math> if and only if there exists <math>\alpha</math> such that <math>(\beta_1,\ldots, \beta_n, \alpha)</math> is a common zero of {{math|''A''}} and {{math|''B''}}. Such a common zero is also a zero of all elements of <math>I\cap R.</math> Conversely, if <math>(\beta_1,\ldots, \beta_n)</math> is a common zero of the elements of <math>I\cap R,</math> it is a zero of the resultant, and there exists <math>\alpha</math> such that <math>(\beta_1,\ldots, \beta_n, \alpha)</math> is a common zero of {{math|''A''}} and {{math|''B''}}. So <math>I\cap R</math> and <math>R\operatorname{res}_x(A,B)</math> have exactly the same zeros.

==Computation==

Theoretically, the resultant could be computed by using the formula expressing it as a product of roots differences. However, as the roots may generally not be computed exactly, such an algorithm would be inefficient and [[numerically unstable]]. As the resultant is a [[symmetric polynomial|symmetric function]] of the roots of each polynomial, it could also be computed by using the [[fundamental theorem of symmetric polynomials]], but this would be highly inefficient.

As the resultant is the [[determinant]] of the [[Sylvester matrix]] (and of the [[Bézout matrix]]), it may be computed by using any algorithm for computing determinants. This needs <math>O(n^3)</math> arithmetic operations. As algorithms are known with a better complexity (see below), this method is not used in practice.

It follows from {{slink||Invariance under change of polynomials}} that the computation of a resultant is strongly related to the [[Polynomial greatest common divisor#Euclid's algorithm|Euclidean algorithm for polynomials]]. This shows that the computation of the resultant of two polynomials of degrees {{math|''d''}} and {{math|''e''}} may be done in <math>O(de)</math> arithmetic operations in the field of coefficients.

However, when the coefficients are integers, rational numbers or polynomials, these arithmetic operations imply a number of GCD computations of coefficients which is of the same order and make the algorithm inefficient.
The [[Polynomial greatest common divisor#Subresultant pseudo-remainder sequence|subresultant pseudo-remainder sequence]]s were introduced to solve this problem and avoid any fraction and any GCD computation of coefficients. A more efficient algorithm is obtained by using the good behavior of the resultant under a ring homomorphism on the coefficients: to compute a resultant of two polynomials with integer coefficients, one computes their resultants modulo sufficiently many [[prime number]]s and then reconstructs the result with the [[Chinese remainder theorem]].

The use of [[fast multiplication]] of integers and polynomials allows algorithms for resultants and greatest common divisors that have a better [[time complexity]], which is of the order of the complexity of the multiplication, multiplied by the logarithm of the size of the input (<math>\log(s(d+e)),</math> where {{math|''s''}} is an upper bound of the number of digits of the input polynomials).

==Application to polynomial systems==

Resultants were introduced for solving [[systems of polynomial equations]] and provide the oldest proof that there exist [[algorithm]]s for solving such systems. These are primarily intended for systems of two equations in two unknowns, but also  allow solving general systems.

===Case of two equations in two unknowns===

Consider the system of two polynomial equations
<math display="block">\begin{align}
P(x,y)&=0\\
Q(x,y)&=0,
\end{align}</math>
where {{math|''P''}} and {{math|''Q''}} are polynomials of respective [[total degree]]s {{math|''d''}} and {{math|''e''}}. Then <math>R=\operatorname{res}_y^{d,e}(P,Q)</math> is a polynomial in {{math|''x''}}, which is [[generic property|generically]] of degree {{math|''de''}} (by properties of {{slink||Homogeneity}}). A value <math>\alpha</math> of {{math|''x''}} is a root of {{math|''R''}} if and only if either there exist <math>\beta</math> in an [[algebraically closed field]] containing the coefficients, such that <math>P(\alpha,\beta) = Q(\alpha,\beta)=0</math>, or <math>\deg(P(\alpha,y)) <d </math> and <math>\deg(Q(\alpha,y)) <e </math> (in this case, one says that {{math|''P''}} and {{math|''Q''}} have a common root at infinity for <math>x=\alpha</math>).

Therefore, solutions to the system are obtained by computing the roots of {{math|''R''}}, and for each root <math>\alpha,</math> computing the common root(s) of <math>P(\alpha,y),</math> <math>Q(\alpha,y),</math> and <math>\operatorname{res}_x(P,Q).</math>

[[Bézout's theorem]] results from the value of <math>\deg\left(\operatorname{res}_y(P,Q)\right)\le de</math>, the product of the degrees of {{math|''P''}} and {{math|''Q''}}. In fact, after a linear change of variables, one may suppose that, for each root {{math|''x''}} of the resultant, there is exactly one value of {{math|''y''}} such that {{math|(''x'', ''y'')}} is a common zero of {{math|''P''}} and {{math|''Q''}}. This shows that the number of common zeros is at most the degree of the resultant, that is at most the product of the degrees of {{math|''P''}} and {{math|''Q''}}. With some technicalities, this proof may be extended to show that, counting multiplicities and zeros at infinity, the number of zeros is exactly the product of the degrees.

===General case===
At first glance, it seems that resultants may be applied to a general [[polynomial system of equations]]
<math display="block">\begin{align}
P_1(x_1, \ldots, x_n) &= 0 \\
&\;\;\vdots \\
P_k(x_1, \ldots, x_n) &= 0
\end{align}</math>
by computing the resultants of every pair <math>(P_i,P_j)</math> with respect to <math>x_n</math> for eliminating one unknown, and repeating the process until getting univariate polynomials. Unfortunately, this introduces many spurious solutions, which are difficult to remove.

A method, introduced at the end of the 19th century{{cn|date=June 2025}}, works as follows: introduce {{math|''k'' − 1}} new indeterminates <math>U_2, \ldots, U_k</math> and compute
<math display="block">\operatorname{res}_{x_n}(P_1, U_2P_2 +\cdots +U_kP_k).</math> 
This is a polynomial in <math>U_2, \ldots, U_k</math> whose coefficients are polynomials in <math>x_1, \ldots, x_{n-1},</math> which have the property that <math>\alpha_1, \ldots, \alpha_{n-1}</math> is a common zero of these polynomial coefficients, if and only if the univariate polynomials <math>P_i(\alpha_1, \ldots, \alpha_{n-1}, x_n)</math> have a common zero, possibly [[point at infinity|at infinity]]. This process may be iterated until finding univariate polynomials.

To get a correct algorithm two complements have to be added to the method. Firstly, at each step, a linear change of variable may be needed in order that the degrees of the polynomials in the last variable are the same as their total degree. Secondly, if, at any step, the resultant is zero, this means that the polynomials have a common factor and that the solutions split in two components: one where the common factor is zero, and the other which is obtained by factoring out this common factor before continuing.

This algorithm is very complicated and has a huge [[time complexity]]{{cn|date=June 2025}}. Therefore, its interest is mainly historical.

==Other applications==

===Number theory===
The [[discriminant]] of a polynomial, which is a fundamental tool in [[number theory]], is <math>a_0^{-1} (-1)^{n(n - 1)/2}\operatorname{res}_x(f(x), f'(x)) </math>, where <math>a_0</math> is the leading coefficient of <math>f(x)</math> and <math>n</math> its degree.

If <math>\alpha</math> and <math>\beta</math> are [[algebraic numbers]] such that <math>P(\alpha) = Q(\beta) = 0</math>, then <math>\gamma = \alpha+\beta</math> is a root of the resultant <math>\operatorname{res}_x(P(x),Q(z-x)),</math> and <math>\tau = \alpha\beta</math> is a root of <math>\operatorname{res}_x(P(x),x^nQ(z/x))</math>, where <math>n</math> is the [[Degree of a polynomial|degree]] of <math>Q(y)</math>. Combined with the fact that <math>1/\beta</math> is a root of <math>y^n Q(1/y) = 0</math>, this shows that the set of algebraic numbers is a [[Field (mathematics)|field]].

Let <math>K(\alpha)</math> be an algebraic field extension generated by an element <math>\alpha,</math> which has <math>P(x)</math> as [[minimal polynomial (field theory)|minimal polynomial]]. Every element of <math>\beta \in K(\alpha)</math> may be written as <math>\beta = Q(\alpha),</math> where <math>Q</math> is a polynomial. Then <math>\beta</math> is a root of <math>\operatorname{res}_x(P(x),z-Q(x)),</math> and this resultant is a power of the minimal polynomial of <math>\beta.</math>

===Algebraic geometry===
Given two [[plane algebraic curve]]s defined as the zeros of the polynomials {{math|''P''(''x'', ''y'')}} and {{math|''Q''(''x'', ''y'')}}, the resultant allows the computation of their intersection. More precisely, the roots of <math>\operatorname{res}_y(P,Q)</math> are the ''x''-coordinates of the intersection points and of the common vertical asymptotes, and the roots of <math>\operatorname{res}_x(P,Q)</math> are the ''y''-coordinates of the intersection points and of the common horizontal asymptotes.

A [[rational curve|rational plane curve]] may be defined by a [[parametric equation]]
<math display="block">
x = \frac{P(t)}{R(t)},\qquad
y = \frac{Q(t)}{R(t)},
</math>
where {{math|''P''}}, {{math|''Q''}} and {{math|''R''}} are polynomials. An [[implicit equation]] of the curve is given by
<math display="block">\operatorname{res}_t(x R - P, y R - Q).</math>
The ''degree'' of this curve is the highest degree of {{math|''P''}}, {{math|''Q''}} and {{math|''R''}}, which is equal to the total degree of the resultant.

===Symbolic integration===
In [[symbolic integration]], for computing the [[antiderivative]] of a [[rational fraction]], one uses [[partial fraction decomposition]] for decomposing the integral into a "rational part", which is a sum of rational fractions whose antiprimitives are rational fractions, and a "logarithmic part" which is a sum of rational fractions of the form
<math display="block">\frac{P(x)}{Q(x)},</math>
where {{math|''Q''}} is a [[square-free polynomial]] and {{math|''P''}} is a polynomial of lower degree than {{math|''Q''}}. The antiderivative of such a function involves necessarily [[logarithms]], and generally algebraic numbers (the roots of {{math|''Q''}}). In fact, the antiderivative is 
<math display="block">\int \frac{P(x)}{Q(x)} dx = \sum_{Q(\alpha)=0} \frac{P(\alpha)}{Q'(\alpha)} \log(x-\alpha),</math>
where the sum runs over all complex roots of {{math|''Q''}}.

The number of [[algebraic number]]s involved by this expression is generally equal to the degree of {{math|''Q''}}, but it occurs frequently that an expression with less algebraic numbers may be computed. The [[Daniel Lazard|Lazard]]–Rioboo–[[Barry Trager|Trager]] method produces an expression, where the number of algebraic numbers is minimal, without any computation with algebraic numbers.

Let 
<math display="block"> S_1(r) S_2(r)^2 \cdots S_k(r)^k = \operatorname{res}_r (rQ'(x)-P(x), Q(x))</math> 
be the [[square-free factorization]] of the resultant which appears on the right. Trager proved that the antiderivative is 
<math display="block">\int \frac{P(x)}{Q(x)}dx = \sum_{i=1}^k\sum_{S_i(\alpha)=0} \alpha \log(T_i(\alpha,x)),</math>
where the internal sums run over the roots of the <math>S_i</math> (if <math>S_i=1</math> the sum is zero, as being the [[empty sum]]), and <math>T_i(r,x)</math> is a polynomial of degree {{math|''i''}} in {{math|''x''}}. The Lazard-Rioboo contribution is the proof that <math>T_i(r,x)</math> is the [[polynomial greatest common divisor#Subresultants|subresultant]] of degree {{math|''i''}} of <math>rQ'(x)-P(x)</math> and <math>Q(x).</math> It is thus obtained for free if the resultant is computed by the [[polynomial greatest common divisor#Subresultant pseudo-remainder sequence|subresultant pseudo-remainder sequence]].

===Computer algebra===
All preceding applications, and many others, show that the resultant is a fundamental tool in [[computer algebra]]. In fact most [[computer algebra systems]] include an efficient implementation of the computation of resultants.

==Homogeneous resultant==

The resultant is also defined for two [[homogeneous polynomial]] in two indeterminates. Given two homogeneous polynomials {{math|''P''(''x'', ''y'')}} and {{math|''Q''(''x'', ''y'')}} of respective [[total degree]]s {{math|''p''}} and {{math|''q''}}, their '''homogeneous resultant''' is the [[determinant]] of the matrix over the [[monomial basis]] of the [[linear map]]
<math display="block">(A,B) \mapsto AP+BQ,</math>
where {{math|''A''}} runs over the bivariate homogeneous polynomials of degree {{math|''q'' − 1}}, and {{math|''B''}} runs over the homogeneous polynomials of degree {{math|''p'' − 1}}. In other words, the homogeneous resultant of {{math|''P''}} and {{math|''Q''}} is the resultant of 
{{math|''P''(''x'', 1)}} and {{math|''Q''(''x'', 1)}} when they are considered as polynomials of degree {{math|''p''}} and {{math|''q''}} (their degree in {{math|''x''}} may be lower than their total degree):
<math display="block">\operatorname{Res}(P(x,y),Q(x,y)) = \operatorname{res}_{p,q}(P(x,1),Q(x,1)). </math>
(The capitalization of "Res" is used here for distinguishing the two resultants, although there is no standard rule for the capitalization of the abbreviation).

The homogeneous resultant has essentially the same properties as the usual resultant, with essentially two differences: instead of polynomial roots, one considers zeros in the [[projective line]], and the degree of a polynomial may not change under a [[ring homomorphism]].
That is:
* The resultant of two homogeneous polynomials over an [[integral domain]] is zero if and only if they have a non-zero common zero over an [[algebraically closed field]] containing the coefficients.
* If {{math|''P''}} and {{math|''Q''}} are two bivariate homogeneous polynomials with coefficients in a [[commutative ring]] {{math|''R''}}, and <math>\varphi\colon R\to S</math> a [[ring homomorphism]] of {{math|''R''}} into another commutative ring {{math|''S''}}, then extending <math>\varphi</math> to polynomials over {{math|''R''}}, ones has <math display="block">\operatorname{Res}(\varphi(P), \varphi(Q)) = \varphi(\operatorname{Res}(P,Q)).</math>
* The property of an homogeneous resultant to be zero is invariant under any projective change of variables.

Any property of the usual resultant may similarly extended to the homogeneous resultant, and the resulting property is either very similar or simpler than the corresponding property of the usual resultant.

==Macaulay's resultant==

'''Macaulay's resultant''', named after [[Francis Sowerby Macaulay]], also called the '''multivariate resultant''', or the '''multipolynomial resultant''',<ref>{{Citation | last1=Cox | first1=David | last2=Little | first2=John | last3=O'Shea | first3=Donal | title=Using Algebraic Geometry | publisher=[[Springer Science+Business Media]] | isbn=978-0387207339 | year=2005}}, Chapter 3. Resultants</ref> is a generalization of the homogeneous resultant to {{math|''n''}} [[homogeneous polynomial]]s in {{math|''n''}} [[indeterminate (variable)|indeterminates]]. Macaulay's resultant is a polynomial in the coefficients of these {{math|''n''}} homogeneous polynomials that vanishes if and only if the polynomials have a common non-zero solution in an [[algebraically closed field]] containing the coefficients, or, equivalently, if the {{math|''n''}} hyper surfaces defined by the polynomials have a common zero in the {{math|''n''  –1}} dimensional projective space. The multivariate resultant is, with [[Gröbner basis|Gröbner bases]], one of the main tools of effective [[elimination theory]] (elimination theory on computers).

Like the homogeneous resultant, Macaulay's may be defined with [[determinants]], and thus behaves well under [[ring homomorphism]]s. However, it cannot be defined by a single determinant. It follows that it is easier to define it first on [[generic polynomial]]s.

===Resultant of generic homogeneous polynomials===
A homogeneous polynomial of degree {{math|''d''}} in {{math|''n''}} variables may have up to 
<math display="block">\binom{n+d-1}{n-1}=\frac{(n+d-1)!}{(n-1)!\,d!}</math>
coefficients; it is said to be ''generic'', if these coefficients are distinct indeterminates.

Let <math>P_1, \ldots, P_n</math> be {{math|''n''}} generic homogeneous polynomials in {{math|''n''}} indeterminates, of respective [[total degree|degrees]] <math>d_1, \dots, d_n.</math> Together, they involve
<math display="block">\sum_{i=1}^n\binom{n+d_i-1}{n-1}</math>
indeterminate coefficients.
Let {{math|''C''}} be the polynomial ring over the integers, in all these 
indeterminate coefficients. The polynomials <math>P_1, \ldots, P_n</math> belong thus to <math>C[x_1,\ldots, x_n],</math> and their resultant (still to be defined) belongs to {{math|''C''}}.

The '''Macaulay degree''' is the integer <math>D=d_1+\cdots+d_n-n+1,</math> which is fundamental in Macaulay's theory. For defining the resultant, one considers the '''Macaulay matrix''', which is the matrix over the [[monomial basis]] of the {{math|''C''}}-linear map
<math display="block">(Q_1, \ldots, Q_n)\mapsto Q_1P_1+\cdots+Q_nP_n,</math>
in which each <math>Q_i</math> runs over the homogeneous polynomials of degree <math>D-d_i,</math> and the [[codomain]] is the {{math|''C''}}-module of the homogeneous polynomials of degree {{math|''D''}}.

If {{math|1=''n'' = 2}}, the Macaulay matrix is the Sylvester matrix, and is a [[square matrix]], but this is no longer true for {{math|''n'' > 2}}. Thus, instead of considering the determinant, one considers all the maximal [[minor (linear algebra)|minors]], that is the determinants of the square submatrices that have as many rows as the Macaulay matrix. Macaulay proved that the {{math|''C''}}-ideal generated by these principal minors is a [[principal ideal]], which is generated by the [[greatest common divisor]] of these minors. As one is working with polynomials with integer coefficients, this greatest common divisor is defined up to its sign. The '''generic Macaulay resultant''' is the greatest common divisor which becomes {{math|1}}, when, for each {{math|''i''}}, zero is substituted for all coefficients of <math>P_i,</math> except the coefficient of <math>x_i^{d_i},</math> for which one is substituted.

====Properties of the generic Macaulay resultant====
*The generic Macaulay resultant is an [[irreducible polynomial]].
*It is homogeneous of degree <math>B/d_i</math> in the coefficients of <math>P_i,</math> where <math>B=d_1 \cdots d_n</math> is the [[Bézout's theorem|Bézout bound]].
*The product with the resultant of every monomial of degree {{math|''D''}} in <math>x_1,\dots, x_n</math> belongs to the ideal of <math>C[x_1,\dots,x_n]</math> generated by <math>P_1,\dots,P_n.</math>

===Resultant of polynomials over a field===
From now on, we consider that the homogeneous polynomials <math>P_1,\ldots,P_n</math> of degrees <math>d_1,\ldots,d_n</math> have their coefficients in a [[field (mathematics)|field]] {{math|''k''}}, that is that they belong to <math>k[x_1,\dots,x_n].</math> Their '''resultant''' is defined as the element of {{math|''k''}} obtained by replacing in the generic resultant the indeterminate coefficients by the actual coefficients of the <math>P_i.</math>

The main property of the resultant is that it is zero if and only if <math>P_1,\ldots,P_n</math> have a nonzero common zero in an [[algebraically closed extension]] of {{math|''k''}}.

The "only if" part of this theorem results from the last property of the preceding paragraph, and is an effective version of [[Hilbert's Nullstellensatz#Projective Nullstellensatz|projective Nullstellensatz]]: If the resultant is nonzero, then
<math display="block">\langle x_1,\ldots, x_n\rangle^D \subseteq \langle P_1,\ldots,P_n\rangle,</math>
where <math>D=d_1+\cdots +d_n-n+1</math> is the Macaulay degree, and <math>\langle x_1,\ldots, x_n\rangle</math> is the maximal homogeneous ideal. This implies that <math>P_1,\ldots,P_n</math> have no other common zero than the unique common zero, {{math|(0, ..., 0)}}, of <math>x_1,\ldots,x_n.</math>

===Computability===
As the computation of a resultant may be reduced to computing determinants and [[polynomial greatest common divisor]]s, there are [[algorithm]]s for computing resultants in a finite number of steps.

However, the generic resultant is a polynomial of very high degree (exponential in {{math|''n''}}) depending on a huge number of indeterminates. It follows that, except for very small {{math|''n''}} and very small degrees of input polynomials, the generic resultant is, in practice, impossible to compute, even with modern computers. Moreover, the number of [[monomial]]s of the generic resultant is so high, that, if it would be computable, the result could not be stored on available memory devices, even for rather small values of {{math|''n''}} and of the degrees of the input polynomials.

Therefore, computing the resultant makes sense only for polynomials whose coefficients belong to a field or are polynomials in few indeterminates over a field.

In the case of input polynomials with coefficients in a field, the exact value of the resultant is rarely important, only its equality (or not) to zero matters. As the resultant is zero if and only if the rank of the Macaulay matrix is lower than its number of its rows, this equality to zero may by tested by applying [[Gaussian elimination]] to the Macaulay matrix. This provides a [[time complexity|computational complexity]] <math>d^{O(n)},</math> where {{math|''d''}} is the maximum degree of input polynomials.

Another case where the computation of the resultant may provide useful information is when the coefficients of the input polynomials are polynomials in a small number of indeterminates, often called parameters. In this case, the resultant, if not zero, defines a [[hypersurface]] in the parameter space. A point belongs to this hyper surface, if and only if there are values of <math>x_1, \ldots,x_n</math> which, together with the coordinates of the point are a zero of the input polynomials. In other words, the resultant is the result of the "[[elimination theory|elimination]]" of <math>x_1, \ldots,x_n</math> from the input polynomials.

===''U''-resultant {{anchor|Uresultant}}===
Macaulay's resultant provides a method, called "''U''-resultant" by Macaulay, for solving [[systems of polynomial equations]].

Given {{math|''n'' − 1}} homogeneous polynomials <math>P_1, \ldots, P_{n-1},</math> of degrees <math>d_1, \ldots, d_{n-1},</math> in  {{math|''n''}} indeterminates <math>x_1, \ldots, x_n,</math> over a field {{math|''k''}},  their '''''U''-resultant''' is the resultant of the {{math|''n''}} polynomials <math>P_1, \ldots, P_{n-1}, P_n,</math> where
<math display="block">P_n = u_1 x_1 + \cdots + u_n x_n</math>
is the generic [[linear form]] whose coefficients are new indeterminates <math>u_1, \ldots, u_n.</math> Notation <math>u_i</math> or <math>U_i</math> for these generic coefficients is traditional, and is the origin of the term ''U''-resultant.

The ''U''-resultant is a homogeneous polynomial in <math>k[u_1, \ldots, u_n].</math> It is zero if and only if the common zeros of <math>P_1, \ldots, P_{n-1}</math> form a [[projective algebraic set]] of positive [[dimension of an algebraic variety|dimension]] (that is, there are infinitely many projective zeros over an  [[algebraically closed extension]] of {{math|''k''}}). If the ''U''-resultant is not zero, its degree is the [[Bézout's theorem|Bézout bound]] <math>d_1\cdots d_{n-1}.</math>
The ''U''-resultant factorizes over an algebraically closed extension of {{math|''k''}} into a product of linear forms. If <math>\alpha_1u_1+\ldots+\alpha_nu_n</math> is such a linear factor, then <math>\alpha_1, \ldots, \alpha_n</math> are the [[homogeneous coordinates]] of a common zero of <math>P_1, \ldots, P_{n-1}.</math> Moreover, every common zero may be obtained from one of these linear factors, and the multiplicity as a factor is equal to the [[intersection multiplicity]] of the <math>P_i</math> at this zero. In other words, the ''U''-resultant provides a completely explicit version of [[Bézout's theorem]].

====Extension to more polynomials and computation====
The ''U''-resultant as defined by Macaulay requires the number of homogeneous polynomials in the system of equations to be  <math>n-1</math>, where  <math>n</math> is the number of indeterminates. In 1981, [[Daniel Lazard]] extended the notion to the case where the number of polynomials may differ from <math>n-1</math>, and the resulting computation can be performed via a specialized [[Gaussian elimination]] procedure followed by symbolic [[determinant]] computation.

Let <math>P_1, \ldots, P_k</math> be homogeneous polynomials in <math>x_1, \ldots, x_n,</math> of degrees <math>d_1, \ldots, d_k,</math> over a field {{math|''k''}}. Without loss of generality, one may suppose that <math>d_1\ge d_2\ge \cdots \ge d_k.</math> Setting <math>d_i=1</math> for {{math|''i'' > ''k''}}, the Macaulay bound is <math>D=d_1+\cdots + d_n-n+1.</math>

Let <math>u_1, \ldots, u_n</math> be new indeterminates and define <math>P_{k+1}=u_1x_1+\cdots +u_nx_n.</math> In this case, the Macaulay matrix is defined to be the matrix, over the basis of the monomials in <math>x_1, \ldots, x_n,</math> of the linear map
<math display="block">(Q_1, \ldots, Q_{k+1}) \mapsto P_1Q_1+\cdots+P_{k+1}Q_{k+1},</math>
where, for each {{math|''i''}}, <math>Q_i</math> runs over the linear space consisting of zero and the homogeneous polynomials of degree <math>D-d_i</math>.

Reducing the Macaulay matrix by a variant of [[Gaussian elimination]], one obtains a square matrix of [[linear form]]s in <math>u_1, \ldots, u_n.</math> The [[determinant]] of this matrix is the ''U''-resultant. As with the original ''U''-resultant, it is zero if and only if <math>P_1, \ldots, P_k</math> have infinitely many common projective zeros (that is if the [[projective algebraic set]] defined by <math>P_1, \ldots, P_k</math> has infinitely many points over an [[algebraic closure]] of {{math|''k''}}). Again as with the original ''U''-resultant, when this ''U''-resultant is not zero, it factorizes into linear factors over any algebraically closed extension of {{math|''k''}}. The coefficients of these linear factors are the [[homogeneous coordinates]] of the common zeros of <math>P_1, \ldots, P_k,</math> and the multiplicity of a common zero equals the multiplicity of the corresponding linear factor.

The number of rows of the Macaulay matrix is less than <math>(ed)^n,</math> where {{math|1=''e'' ~ 2.7182}} is the usual [[e (mathematical constant)|mathematical constant]], and {{math|''d''}} is the [[arithmetic mean]] of the degrees of the <math>P_i.</math> It follows that all solutions of a [[system of polynomial equations]] with a finite number of projective zeros can be determined in [[time complexity|time]] <math>d^{O(n)}.</math> Although this bound is large, it is nearly optimal in the following sense: if all input degrees are equal, then the time complexity of the procedure is polynomial in the expected number of solutions ([[Bézout's theorem]]). This computation may be practically viable when  {{math|''n''}}, {{math|''k''}} and {{math|''d''}} are not large.

==See also==
*[[Elimination theory]]
*[[Subresultant]]
*[[Nonlinear algebra]]

==Notes==
{{notelist}}

==References==

{{Reflist}}
 
{{refbegin}}
*{{Citation | last1=Gelfand | first1=I. M. | last2=Kapranov | first2=M.M. | last3=Zelevinsky | first3=A.V. | title=Discriminants, resultants, and multidimensional determinants | publisher=Boston: Birkhäuser | isbn=978-0-8176-3660-9 | year=1994}}
*{{citation|first=F. S. |last=Macaulay|author-link=Francis Sowerby Macaulay|title=The Algebraic Theory of Modular Systems|url= https://archive.org/details/algebraictheoryo00macauoft| year=1916|isbn= 978-1275570412| series=The Cornell Library of Historical Mathematical Monographs|publisher=Cambridge University Press}}
{{refend}}

==External links==
* {{MathWorld |urlname=Resultant |title=Resultant}}

{{Polynomials}}

[[Category:Polynomials]]
[[Category:Determinants]]
[[Category:Computer algebra]]