Resultant

Notation

The resultant of two univariate polynomials A and B is commonly denoted res ⁡ ( A , B ) or Res ⁡ ( A , B ) .

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: res x ⁡ ( A , B ) or Res x ⁡ ( A , B ) .

The degree of the polynomials are used in the definition of the resultant. However, a polynomial of degree d may also be considered as a polynomial of higher degree such 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 res d , e ⁡ ( A , B ) or res x d , e ⁡ ( A , B ) .

Definition

The resultant of two univariate polynomials over a field or over a commutative ring is commonly defined as the determinant of their Sylvester matrix. More precisely, let

A = a 0 x d + a 1 x d − 1 + ⋯ + a d

and

B = b 0 x e + b 1 x e − 1 + ⋯ + b e

be nonzero polynomials of respective degrees d and e. Let us denote by P i the vector space (or free module if the coefficients belong to a commutative ring) of dimension i whose elements are the polynomials of degree less than i. The map

φ : P e × P d → P d + e

such that

φ ( P , Q ) = A P + B Q

is a linear map between two spaces of the same dimension. Over the basis of the powers of x, this map is represented by a square matrix of dimension d + e, which called the Sylvester matrix of A and 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 A and B is thus the determinant

| a 0 0 ⋯ 0 b 0 0 ⋯ 0 a 1 a 0 ⋯ 0 b 1 b 0 ⋯ 0 a 2 a 1 ⋱ 0 b 2 b 1 ⋱ 0 ⋮ ⋮ ⋱ a 0 ⋮ ⋮ ⋱ b 0 ⋮ ⋮ ⋯ a 1 ⋮ ⋮ ⋯ b 1 a d a d − 1 ⋯ ⋮ b e b e − 1 ⋯ ⋮ 0 a d ⋱ ⋮ 0 b e ⋱ ⋮ ⋮ ⋮ ⋱ a d − 1 ⋮ ⋮ ⋱ b e − 1 0 0 ⋯ a d 0 0 ⋯ b e | ,

which has e columns of a_i and d columns of b_j (for simplification, d = e in the displayed determinant).

In the case of monic polynomials over an integral domain the resultant is equal to the product

∏ ( x , y ) : f ( x ) = g ( y ) = 0 ( x − y ) ,

where x and y run over the roots of the polynomials over an algebraically closed field containing the coefficients. For non-monic polynomials with leading coefficients a₀ and b₀ , respectively, the above product is multiplied by a 0 e b 0 d .

Properties

In this section and its subsections, A and B are two polynomials in x of respective degrees d and e, and their resultant is denoted res ⁡ ( A , B ) .

Characterizing properties

If d = 0 (that is if A = a 0 is a nonzero constant) then res ⁡ ( A , B ) = a 0 e . Similarly, if e = 0, then res ⁡ ( A , B ) = b 0 d .

res ⁡ ( x − a 1 , x − b 1 ) = b 1 − a 1

res ⁡ ( B , A ) = ( − 1 ) d e res ⁡ ( A , B )

res ⁡ ( A B , C ) = res ⁡ ( A , C ) res ⁡ ( B , C )

The preceding properties characterize the resultant. In other words, the resultant is the unique function of the coefficients of polynomials that has these properties.

Some mathematics software, including Mathematica and Maple, use the alternate convention res ⁡ ( x − a 1 , x − b 1 ) = a 1 − b 1 .

Zeros

The resultant of two polynomials with coefficients in an integral domain 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 exist a polynomial P of degree less than e and a polynomial Q of degree less than d such that res ⁡ ( A , B ) = A P + B Q . 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 generated by these polynomials.

Invariance by ring homomorphisms

Let A and B be two polynomials of respective degrees d and e with coefficients in a commutative ring R, and φ : R → S a ring homomorphism of R into another commutative ring S. Applying φ to the coefficients of a polynomial extends φ to a homomorphism of polynomial rings R [ x ] → S [ x ] , which is also denoted φ . With this notation, we have:

If φ preserve the degrees of A and B (that is if deg ⁡ ( φ ( A ) ) = d and deg ⁡ ( φ ( B ) ) = e ), then

If deg ⁡ ( φ ( A ) ) < d and deg ⁡ ( φ ( B ) ) < e , then

If deg ⁡ ( φ ( A ) ) = d and deg ⁡ ( φ ( B ) ) = f < e , and the leading coefficient of A is a 0 then

If deg ⁡ ( φ ( A ) ) = f < d and deg ⁡ ( φ ( B ) ) = e , and the leading coefficient of B is b 0 then

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 modulo several primes and to retrieve the desired resultant with Chinese remainder theorem. When R is a polynomial ring in other indeterminates, and S is the ring obtained by specializing to numerical values some or all indeterminates of 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

res ⁡ ( A ( x + a ) , B ( x + a ) ) = res ⁡ ( A ( x ) , B ( x ) )

res ⁡ ( A ( a x ) , B ( a x ) ) = a d e res ⁡ ( A ( x ) , B ( x ) )

If A r ( x ) = x d A ( 1 / x ) and B r ( x ) = x e B ( 1 / x ) are the reciprocal polynomials of A and B, respectively, then

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 a and b are nonzero constants (that is they are independent of the indeterminate x), and A and B are as above, then

If d = deg ⁡ ( A ) ≥ e = deg ⁡ ( B ) , if a is a constant and b 0 is the leading coefficient of B, and if C is a polynomial of degree at most d − e then

These properties imply that in Euclidean algorithm for polynomials, the resultant of two successive remainders differs from the resultant of the initial polynomials by a factor, which is easy to compute. Moreover, the constant a in above second formula may be chosen in order that the successive remainders have their coefficients in the ring of coefficients of input polynomials. This is the starting idea of the subresultant-pseudo-remainder-sequence algorithm for computing the greatest common divisor and the resultant of two polynomials. This algorithms works for polynomials over the integers or, more generally, over an integral domain, without any other division than exact divisions (that is without involving fractions). It involves O ( d e ) arithmetic operations, while the computation of the determinant of the Sylvester matrix with standard algorithms require O ( ( d + e ) 3 ) arithmetic operations.

Generic properties

In this section, we consider two polynomials

A = a 0 x d + a 1 x d − 1 + ⋯ + a d

and

B = b 0 x e + b 1 x e − 1 + ⋯ + b e

whose d + e + 2 coefficients are distinct indeterminates. Let

R = Z [ a 0 , … , a d , b 0 , … , b e ]

be the polynomial ring over the integers defined by these indeterminates. The resultant res ⁡ ( A , B ) is often called the generic resultant for the degrees d and e. It has the following properties.

res ⁡ ( A , B ) is an absolutely irreducible polynomial.

If I is the ideal of R [ x ] generated by A and B, then I ∩ R is the principal ideal generated by res ⁡ ( A , B ) .

Homogeneity

The generic resultant for the degrees d and e is homogeneous in various ways. More precisely:

It is homogeneous of degree e in a 0 , … , a d .

It is homogeneous of degree d in b 0 , … , b e .

It is homogeneous of degree d + e in all the variables a i and b j .

If a i and b i are given the weight i (that is, the weight of each coefficient is its degree as elementary symmetric polynomial), then it is quasi-homogeneous of total weight de.

If P and Q are homogeneous multivariate polynomials of respective degrees d and e, then their resultant in degrees d and e with respect to an indeterminate x, denoted res x d , e ⁡ ( P , Q ) in § Notation, is homogeneous of degree de in the other indeterminates.

Elimination properties

Let I = ⟨ A , B ⟩ be the ideal generated by two polynomials A and B in a polynomial ring R [ x ] , where R is itself a polynomial ring over a field. Then:

I ∩ R = R ⋅ r is a principal ideal generated by r for some r ∈ R

res x ⁡ ( A , B ) ∈ R ⋅ r

There exists a positive integer k such that r k ∈ R ⋅ res x ⁡ ( A , B ) .

An example where k > 1 in the latter property is R = R [ y ] , A = x 2 + y 2 − 1 (the unit circle), and B = y − 2. In this case, res x ⁡ ( A , B ) = ( y − 2 ) 2 and ⟨ A , B ⟩ ∩ R = R ⋅ ( y − 2 ) . This example has been chosen for having a prime ideal ⟨ A , B ⟩ . For another field of coefficients and another constant term in B, one has also k > 1, but the ideal may be non-prime.

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 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 O ( n 3 ) arithmetic operations. As one knows algorithms with a better complexity (see below), this method is not used in practice.

It follows from § Invariance under change of polynomials that the computation of a resultant is strongly related with Euclidean algorithm for polynomials. This shows that the computation of the resultant of two polynomials of degrees d and e may be done in O ( d e ) 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 subresultant pseudo-remainder sequences 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 numbers 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 ( log ⁡ ( s ( d + e ) ) , where 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 provides the oldest proof that there exist algorithms for solving such systems. There are primarily intended for systems of two equations in two unknown, but allow also solving general systems.

Case of two equations in two unknowns

Let us consider two polynomials the system of equations

P ( x , y ) = 0 Q ( x , y ) = 0 ,

where P and Q are polynomials of respective total degrees d and e. Then R = res y d , e ⁡ ( P , Q ) is a polynomial in x, which is generically of degree de (by properties of § Homogeneity). A value α of x is a root of R, if and only if, either there exist β in an algebraically closed field containing the coefficients, such that P ( α , β ) = Q ( α , β ) = 0 , or deg ⁡ ( P ( α , y ) ) < d and deg ⁡ ( Q ( α , y ) ) < d (in this case, one says that P and Q have a common root at infinity for x = α ).

Therefore, solving the system amounts computing the roots of R, and for each root α , computing the common root(s) of P ( α , y ) , Q ( α , y ) , and res x ⁡ ( P , Q ) .

It is worth to remark that Bézout's theorem results of the value of deg ⁡ ( res y d , e ⁡ ( P , Q ) ) .

General case

At first glance, it seems that resultants may be applied to a general polynomial system of equations

P 1 ( x 1 , … , x n ) = 0 ⋮ P k ( x 1 , … , x n ) = 0

by computing the resultants of every pair ( P i , P j ) with respect to x n for eliminating one unknown, and repeating the process until getting univariate polynomials. Unfortunately, this introduce many spurious solutions, which are difficult to remove.

A method, introduced at the end of 19th century, works as follows: introduce k − 1 new indeterminates U 2 , … , U k and compute

res x n ⁡ ( P 1 , U 2 P 2 + ⋯ + U k P k ) .

This is a polynomial in U 2 , … , U k whose coefficients are polynomials in x 1 , … , x n − 1 , which have the property that α 1 , … , α n − 1 is a common zero of these polynomial coefficients, if and only if the univariate polynomials P i ( α 1 , … , α n − 1 , x n ) have a common zero, possibly at infinity. This process may be iterated until finding univariate polynomials.

For getting 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, were 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. Therefore, its interest is mainly historical.

Number theory

The discriminant of a polynomial, which is a fundamental tool in number theory is the quotient by its leading coefficient of the resultant of the polynomial and its derivative.

If x and y are algebraic numbers such that P ( x ) = Q ( y ) = 0 , with degree Q of degree n), then z = x + y is a root of the resultant res x ⁡ ( P ( x ) , Q ( z − x ) ) , and t = x y is a root of res x ⁡ ( P ( x ) , x n Q ( t / x ) ) . Combined with the fact that 1 / y is a root of y n Q ( 1 / y ) , this shows that the set of algebraic numbers is a field.

Let K ( α ) be an algebraic field extension generated by an element α , which has P ( x ) as minimal polynomial. Every element of β ∈ K ( α ) may be written as β = Q ( α ) , where Q is a polynomial. Then β is a root of res x ⁡ ( P ( x ) , z − Q ( x ) ) , and this resultant is a power of the minimal polynomial of β .

Algebraic geometry

Given two plane algebraic curves defined as the zeros of the polynomials P(x, y) and Q(x, y), the resultant allows the computation of their intersection. More precisely, the roots of res y ⁡ ( P , Q ) are the x-coordinates of the intersection points and of the common vertical asymptotes, and the roots of res x ⁡ ( P , Q ) are the y-coordinates of the intersection points and of the common horizontal asymptotes.

A rational plane curve may be defined by a parametric equation

x = P ( t ) R ( t ) , y = Q ( t ) R ( t ) ,

where P, Q and R are polynomials. An implicit equation of the curve is given by

res t ⁡ ( x R − P , y R − Q ) .

The degree of this curve is the highest degree of P, Q and 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

P ( x ) Q ( x ) ,

where Q is a square-free polynomial and P is a polynomial of lower degree than Q. The antiderivative of such a function involves necessarily logarithms, and generally algebraic numbers (the roots of Q). In fact, the antiderivative is

∫ P ( x ) Q ( x ) d x = ∑ Q ( α ) = 0 P ( α ) Q ′ ( α ) log ⁡ ( x − α ) ,

where the sum runs over all complex roots of Q.

The number of algebraic numbers involved by this expression is generally equal to the degree of Q, but it occurs frequently that an expression with less algebraic numbers may be computed. The Lazard–Rioboo–Trager method produced an expression, where the number of algebraic numbers is minimal, without any computation with algebraic numbers.

Let

S 1 ( r ) S 2 ( r ) 2 ⋯ S k ( r ) k = res r ⁡ ( r Q ′ ( x ) − P ( x ) , Q ( x ) )

be the square-free factorization of the resultant which appears on the right. Trager proved that the antiderivative is

∫ P ( x ) Q ( x ) d x = ∑ i = 1 k ∑ S i ( α ) = 0 α log ⁡ ( T i ( α , x ) ) ,

where the internal sums run over the roots of the S i (if S i = 1 ) the sum is zero, as being the empty sum), and T i ( r , x ) is a polynomial of degree i in x. The Lazard-Rioboo contribution is the proof that T i ( r , x ) is the subresultant of degree i of r Q ′ ( x ) − P ( x ) and Q ( x ) . It is thus obtained for free if the resultant is computed by the 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 P(x, y) and Q(x, y) of respective total degrees p and q, their homogeneous resultant is the determinant of the matrix over the monomial basis of the linear map

( A , B ) ↦ A P + B Q ,

where A runs over the bivariate homogeneous polynomials of degree q − 1, and B runs over the homogeneous polynomials of degree p − 1. In other words, the homogeneous resultant of P and Q is the resultant of P(x, 1) and Q(x, 1 when they are considered as polynomials of degree p and q (their degree in x may be lower than their total degree):

Res ⁡ ( P ( x , y ) , Q ( x , y ) ) = res p , q ⁡ ( P ( x , 1 ) , Q ( x , 1 ) ) .

(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 P and Q are two bivariate homogeneous polynomials with coefficients in a commutative ring R, and φ : R → S a ring homomorphism of R into another commutative ring S, then extending φ to polynomials over R, ones has

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, is a generalization of the homogeneous resultant to n homogeneous polynomials in n indeterminates. Macaulay's resultant is a polynomial in the coefficients of these 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 n hyper surfaces defined by the polynomials have a common zero in the n –1 dimensional projective space. The multivariate resultant is, with 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 homomorphisms. However, it cannot be defined by a single determinant. It follows that it is easier to define it first on generic polynomials.

Resultant of generic homogeneous polynomials

A homogeneous polynomial of degree d in n variables may have up to

( n + d − 1 n − 1 ) = ( n + d − 1 ) ! ( n − 1 ) ! d !

coefficients; it is said to be generic, if these coefficients are distinct indeterminates.

Let P 1 , … , P n be n generic homogeneous polynomials in n indeterminates, of respective degrees d 1 , … , d n . Together, they involve

∑ i = 1 n ( n + d i − 1 n − 1 )

indeterminate coefficients. Let C be the polynomial ring over the integers, in all these indeterminate coefficients. The polynomials P 1 , … , P n belong thus to C [ x 1 , … , x n ] , and their resultant (still to be defined) belongs to C.

The Macaulay degree is the integer D = d 1 + ⋯ + d n − n + 1 , 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 C-linear map

( Q 1 , … , Q n ) ↦ Q 1 P 1 + ⋯ + Q n P n ,

in which each Q i runs over the homogeneous polynomials of degree D − d i , and the codomain is the C-module of the homogeneous polynomials of degree D.

If n = 2, the Macaulay matrix is the Sylvester matrix, and is a square matrix, but this is no longer true for n > 2. Thus, instead of considering the determinant, one considers all the maximal minors, that is the determinants of the square submatrices that have as many rows as the Macaulay matrix. Macaulay proved that the 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 its sign. The generic Macaulay resultant is the greatest common divisor which becomes 1, when, for each i, zero is substituted for all coefficients of P i , except the coefficient of x i d i , for which one is substituted.

Properties of the generic Macaulay resultant

The generic Macaulay resultant is an irreducible polynomial.

It is homogeneous of degree B / d i in the coefficients of P i , where B = d 1 ⋯ d n is the Bézout bound.

The product with the resultant of every monomial of degree D in x 1 , ⋯ , x n belongs to the ideal of C [ x 1 , ⋯ , x n ] generated by P 1 , … , P n .

Resultant of polynomials over a field

From now on, we consider that the homogeneous polynomials P 1 , … , P n , of degrees d 1 , … , d n have their coefficients in a field k, that is that they belong to k [ x 1 , ⋯ , x n ] . Their resultant is defined as the element of k obtained by replacing in the generic resultant the indeterminate coefficients by the actual coefficients of the P i .

The main property of the resultant is that it is zero if only if P 1 , … , P n , have a nonzero common zero in an algebraically closed extension of k.

The "only if" part of this theorem results on the last property of the preceding paragraph, and is an effective version of Projective Nullstellensatz: If the resultant is nonzerozero, then

⟨ x 1 , … x n ⟩ D ⊆ ⟨ P 1 , … , P n ⟩ ,

where D = d 1 + ⋯ + d n − n + 1 is the Macaulay degree, and ⟨ x 1 , … x n ⟩ is the maximal homogeneous ideal. This implies that P 1 , … , P n have no other common zero that the unique common zero, (0, ..., 0), of x 1 , … , x n .

Computability

As the computation of a resultant may reduced to computing determinants and polynomial greatest common divisors, there are algorithms for computing resultants in a finite number of steps.

However, the generic resultant is a polynomial of very high degree (exponential in n) depending on a huge number of indeterminates. It follows that, except for very small 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 monomials 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 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 computational complexity d O ( n ) , where 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 x 1 , … , x n 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" of x 1 , … , x n from the input polynomials.

U-resultant

Macaulay's resultant provides a method, called "U-resultant" by Macaulay, for solving systems of polynomial equations.

Given n − 1 homogeneous polynomials P 1 , … , P n − 1 , of degrees d 1 , … , d n − 1 , in n indeterminates x 1 , … , x n , over a field k, their U-resultant is the resultant of the n polynomials P 1 , … , P n − 1 , P n , where

P n = u 1 x 1 + ⋯ + u n x n

is the generic linear form whose coefficients are new indeterminates u 1 , … , u n . Notation u i or U i for these generic coefficients is traditional, and is the origin of the term U-resultant.

The U-resultant is a homogeneous polynomial in k [ u 1 , … , u n ] . It is zero if and only if the common zeros of P 1 , … , P n − 1 form an projective algebraic set of positive dimension (that is, there are infinitely many projective zeros over an algebraically closed extension of k). If the U-resultant is not zero, its degree is the Bézout bound d 1 ⋯ d n − 1 . The U-resultant factorizes over an algebraically closed extension of k into a product of linear forms. If α 1 u 1 + … + α n u n is such a linear factor, then α 1 , … , α n are the homogeneous coordinates of a common zero of P 1 , … , P n − 1 . 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 P i 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 was defined only for a number of homogeneous polynomials, which is one less than the number of indeterminates. In 1981, Daniel Lazard provided the following generalization to any number of polynomials, which may be computed by a single Gaussian elimination.

Let P 1 , … , P k be homogeneous polynomials in x 1 , … , x n , of degrees d 1 , … , d k , over a field k. Without generality, one may suppose that d 2 ≥ d 3 ≥ ⋯ ≥ d k ≥ d 1 . Setting d i = 1 for i > k, the Macaulay bound is D = d 1 + ⋯ + d n − n + 1.

Let u 1 , … , u n be new indeterninates, and P k + 1 = u 1 x 1 + ⋯ + u n x n . In this case, the Macaulay matrix is the matrix, over the basis of the monomials in x 1 , … , x n , of the linear map

( Q 1 , … , Q k + 1 ) ↦ P 1 Q 1 + ⋯ + P k + 1 Q k + 1 ,

where, for each i, Q i runs over the homogeneous polynomials of degree D − d i .

Reducing the Macaulay matrix by a variant of Gaussian elimination, one gets a square matrix of linear forms in u 1 , … , u n . The determinant of this matrix is the U-resultant. As for the original U-resultant, it is zero if and only if P 1 , … , P k have infinitely many common projective zeros (that is if the projective algebraic set defined by P 1 , … , P k has infinitely many points over an algebraic closure of k). As for the original U-resultant, when this U-resultant is not zero, it factorizes into linear factors over any algebraically closed extension of k. The coefficients of these linear factors are the homogeneous coordinates of the common zeros of P 1 , … , P k , 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 ( e d ) n , where e ~ 2.7182 is the usual mathematical constant, and d is the arithmetic mean of the degrees of the P i . It follows that, if k is not too large (say, if k ≤ 2n), a system of polynomial equations, that has a finite number of projective zeros, may be completely solved in time d O ( n ) .

Although large, this bound is almost optimal. In fact, if all input degrees are equal, this implies that the time complexity is polynomial in the expected number of solutions (Bézout's theorem).