Polynomial greatest common divisor

To find the GCD of two polynomials using factoring, simply factor the two polynomials completely. Then, take the product of all common factors. At this stage, we do not necessarily have a monic polynomial, so finally multiply this by a constant to make it a monic polynomial. This will be the GCD of the two polynomials as it includes all common divisors and is monic.

Example one: Find the GCD of x² + 7x + 6 and x² − 5x − 6.

x² + 7x + 6 = (x + 1)(x + 6)

x² − 5x − 6 = (x + 1)(x − 6)

Thus, their GCD is x + 1.

Factoring polynomials can be difficult, especially if the polynomials have a large degree. The Euclidean algorithm is a method that works for any pair of polynomials. It makes repeated use of Euclidean division. When using this algorithm on two numbers, the size of the numbers decreases at each stage. With polynomials, the degree of the polynomials decreases at each stage. The last nonzero remainder, made monic if necessary, is the GCD of the two polynomials.

More specifically, for finding the gcd of two polynomials a(x) and b(x), one can suppose b ≠ 0 (otherwise, the GCD is a(x)), and

${\displaystyle \deg(b(x))\leq \deg(a(x))\,.}$

The Euclidean division provides two polynomials q(x), the quotient and r(x), the remainder such that

${\displaystyle a(x)=q_{0}(x)b(x)+r_{0}(x)\qquad {\text{and}}\qquad \deg(r_{0}(x))<\deg(b(x))}$

A polynomial g(x) divides both a(x) and b(x) if and only if it divides both b(x) and r₀(x). Thus

${\displaystyle \gcd(a(x),b(x))=\gcd(b(x),r_{0}(x)).}$

Setting

${\displaystyle a_{1}(x)=b(x),b_{1}(x)=r_{0}(x),}$

one can repeat the Euclidean division to get new polynomials q₁(x), r₁(x), a₂(x), b₂(x) and so on. At each stage we have

${\displaystyle \deg(a_{k+1})+\deg(b_{k+1})<\deg(a_{k})+\deg(b_{k}),}$

so the sequence will eventually reach a point at which

${\displaystyle b_{N}(x)=0}$

and one has got the GCD:

${\displaystyle \gcd(a,b)=\gcd(a_{1},b_{1})=\cdots =\gcd(a_{N},0)=a_{N}.}$

Example: finding the GCD of x² + 7x + 6 and x² − 5x − 6:

x² + 7x + 6 = 1 ⋅ (x² − 5x − 6) + (12 x + 12)

x² − 5x − 6 = (12 x + 12) (1/12 x − 1/2) + 0

Since 12 x + 12 is the last nonzero remainder, it is a GCD of the original polynomials, and the monic GCD is x + 1.

In this example, it is not difficult to avoid introducing denominators by factoring out 12 before the second step. This can always be done by using pseudo-remainder sequences, but, without care, this may introduce very large integers during the computation. Therefore, for computer computation, other algorithms are used, that are described below.

This method works only if one can test the equality to zero of the coefficients that occur during the computation. So, in practice, the coefficients must be integers, rational numbers, elements of a finite field, or must belong to some finitely generated field extension of one of the preceding fields. If the coefficients are floating-point numbers that represent real numbers that are known only approximately, then one must know the degree of the GCD for having a well defined computation result (that is a numerically stable result; in this cases other techniques may be used, usually based on singular value decomposition.

Euclidean division of polynomials, which is used in Euclid's algorithm for computing GCDs, is very similar to Euclidean division of integers. Its existence is based on the following theorem: Given two univariate polynomials a and b ≠ 0 defined over a field, there exist two polynomials q (the quotient) and r (the remainder) which satisfy

${\displaystyle a=bq+r}$

and

${\displaystyle \deg(r)<\deg(b),}$

where "deg(...)" denotes the degree and the degree of the zero polynomial is defined as being negative. Moreover, q and r are uniquely defined by these relations.

The difference from Euclidean division of the integers is that, for the integers, the degree is replaced by the absolute value, and that to have uniqueness one has to suppose that r is non-negative. The rings for which such a theorem exists are called Euclidean domains.

Like for the integers, the Euclidean division of the polynomials may be computed by the long division algorithm. This algorithm is usually presented for paper-and-pencil computation, but it works well on computers when formalized as follows (note that the names of the variables correspond exactly to the regions of the paper sheet in a pencil-and-paper computation of long division). In the following computation "deg" stands for the degree of its argument (with the convention deg(0) < 0), and "lc" stands for the leading coefficient, the coefficient of the highest degree of the variable.

'''Euclidean division'''

'''''Input:''''' ''a'' and ''b'' ≠ 0 two polynomials in the variable ''x'';
'''''Output:''''' ''q'', the quotient, and ''r'', the remainder;

'''''Begin'''''
    ''q'' := 0
    ''r'' := ''a''
    ''d'' := deg(''b'')
    ''c'' := lc(''b'')
    '''''while''''' deg(''r'') ≥ ''d'' '''''do'''''
        ''s'' := <templatestyles src="Sfrac/styles.css" /><span role="math" class="sfrac tion"><span class="num">lc(''r'')</span><span class="sr-only">/</span><span class="den">''c''</span></span> ''x''<sup>deg(''r'')−''d''</sup>
        ''q'' := ''q'' + ''s''
        ''r'' := ''r'' − ''sb''
    '''''end do'''''
    '''''return''''' (''q'', ''r'')
'''''end'''''

The proof of the validity of this algorithm relies on the fact that during the whole "while" loop, we have a = bq + r and deg(r) is a non-negative integer that decreases at each iteration. Thus the proof of the validity of this algorithm also proves the validity of the Euclidean division.

As for the integers, the Euclidean division allows us to define Euclid's algorithm for computing GCDs.

Starting from two polynomials a and b, Euclid's algorithm consists of recursively replacing the pair (a, b) by (b, rem(a, b)) (where "rem(a, b)" denotes the remainder of the Euclidean division, computed by the algorithm of the preceding section), until b = 0. The GCD is the last non zero remainder.

Euclid's algorithm may be formalized in the recursive programming style as:

${\displaystyle \gcd(a,b):={\text{if }}b=0{\text{ then }}a{\text{ else }}\gcd(b,\operatorname {rem} (a,b)).}$

In the imperative programming style, the same algorithm becomes, giving a name to each intermediate remainder:

<span class="nowrap"><math>\begin{align} r_0:=a\\ r_1:=b \end{align}</math></span>

<span class="nowrap">'''''for''''' (<math>i:=1</math>; <math>r_i \neq 0</math>; <math>i:=i+1</math>) '''''do'''''</span>
    <span class="nowrap"><math>\begin{align} r_{i+1}&:=\operatorname{rem}(r_{i-1},r_i) \end{align}</math></span>
'''''end do'''''

<span class="nowrap">'''''return''''' <math>(r_{i-1}).</math></span>

The sequence of the degrees of the r_i is strictly decreasing. Thus after, at most, deg(b) steps, one get a null remainder, say r_k. As (a, b) and (b, rem(a,b)) have the same divisors, the set of the common divisors is not changed by Euclid's algorithm and thus all pairs (r_i, r_i+1) have the same set of common divisors. The common divisors of a and b are thus the common divisors of r_k−1 and 0. Thus r_k−1 is a GCD of a and b. This not only proves that Euclid's algorithm computes GCDs but also proves that GCDs exist.

Bézout's identity is a GCD related theorem, initially proved for the integers, which is valid for every principal ideal domain. In the case of the univariate polynomials over a field, it may be stated as follows.

If g is the greatest common divisor of two polynomials a and b (not both zero), then there are two polynomials u and v such that

${\displaystyle au+bv=g\quad }$ (Bézout's identity)

and either u = 1, v = 0, or u = 0, v = 1, or

${\displaystyle \deg(u)<\deg(b)-\deg(g),\quad \deg(v)<\deg(a)-\deg(g).}$

The interest of this result in the case of the polynomials is that there is an efficient algorithm to compute the polynomials u and v, This algorithm differs from Euclid's algorithm by a few more computations done at each iteration of the loop. It is therefore called extended GCD algorithm. Another difference with Euclid's algorithm is that it also uses the quotient, denoted "quo", of the Euclidean division instead of only the remainder. This algorithm works as follows.

'''Extended GCD''' algorithm

'''''Input:''''' <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>, univariate polynomials

'''''Output:'''''
    <span class="texhtml mvar" style="font-style:italic;">g</span>, the GCD of <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span>
    <span class="texhtml mvar" style="font-style:italic;">u</span>, <span class="texhtml mvar" style="font-style:italic;">v</span>, as in above statement
    <span class="texhtml " >''a''<sub>1</sub></span>, <span class="texhtml " >''b''<sub>1</sub></span>, such that
        <span class="nowrap"><math>\begin{align}
        a=ga_1\\
        b=gb_1
        \end{align}</math></span>
'''''Begin'''''
    <span class="nowrap"><math>\begin{array}{ll}
    r_0  =a & r_1  =b\\
    s_0  =1 & s_1  =0\\
    t_0  =0 & t_1  =1
    \end{array}</math></span>
  
    '''''for''''' (<span class="texhtml " >''i'' = 1</span>; <span class="texhtml " >''r<sub>i</sub>'' ≠ 0</span>; <span class="texhtml " >''i'' = ''i''+1</span>) '''''do'''''
        <span class="nowrap"><math>q  =\operatorname{quo}(r_{i-1},r_{i})</math></span>
        <span class="nowrap"><math>\begin{align}
        r_{i+1} & =r_{i-1}-qr_{i}\\
        s_{i+1} & =s_{i-1}-qs_{i}\\
        t_{i+1} & =t_{i-1}-qt_{i}
        \end{align}</math></span>
    '''''end do'''''
    <span class="nowrap"><math>\begin{align}
    g & =r_{i-1}\\
    u & =s_{i-1}\\
    v & =t_{i-1}
    \end{align}</math></span>
    <span class="nowrap"><math>\begin{align}
    a_1 & =(-1)^{i-1}t_i\\
    b_1 & =(-1)^i s_i
    \end{align}</math></span>
'''''end'''''

The proof that the algorithm satisfies its output specification relies on the fact that, for every i we have

${\displaystyle r_{i}=as_{i}+bt_{i}}$

${\displaystyle s_{i}t_{i+1}-t_{i}s_{i+1}=s_{i}t_{i-1}-t_{i}s_{i-1},}$

the latter equality implying

${\displaystyle s_{i}t_{i+1}-t_{i}s_{i+1}=(-1)^{i}.}$

The assertion on the degrees follows from the fact that, at every iteration, the degrees of s_i and t_i increase at most as the degree of r_i decreases.

An interesting feature of this algorithm is that, when the coefficients of Bezout's identity are needed, one gets for free the quotient of the input polynomials by their GCD.

An important application of the extended GCD algorithm is that it allows one to compute division in algebraic field extensions.

Let L an algebraic extension of a field K, generated by an element whose minimal polynomial f has degree n. The elements of L are usually represented by univariate polynomials over K of degree less than n.

The addition in L is simply the addition of polynomials:

${\displaystyle a+_{L}b=a+_{K[X]}b.}$

The multiplication in L is the multiplication of polynomials followed by the division by f:

${\displaystyle a\cdot _{L}b=\operatorname {rem} (a._{K[X]}b,f).}$

The inverse of a non zero element a of L is the coefficient u in Bézout's identity au + fv = 1, which may be computed by extended GCD algorithm. (the GCD is 1 because the minimal polynomial f is irreducible). The degrees inequality in the specification of extended GCD algorithm shows that a further division by f is not needed to get deg(u) < deg(f).

In the case of univariate polynomials, there is a strong relationship between the greatest common divisors and resultants. More precisely, the resultant of two polynomials P, Q is a polynomial function of the coefficients of P and Q which has the value zero if and only if the GCD of P and Q is not constant.

The subresultants theory is a generalization of this property that allows characterizing generically the GCD of two polynomials, and the resultant is the 0-th subresultant polynomial.[1]

The i-th subresultant polynomial S_i(P ,Q) of two polynomials P and Q is a polynomial of degree at most i whose coefficients are polynomial functions of the coefficients of P and Q, and the i-th principal subresultant coefficient s_i(P ,Q) is the coefficient of degree i of S_i(P, Q). They have the property that the GCD of P and Q has a degree d if and only if

${\displaystyle s_{0}(P,Q)=\cdots =s_{d-1}(P,Q)=0\ ,s_{d}(P,Q)\neq 0}$.

In this case, S_d(P ,Q) is a GCD of P and Q and

${\displaystyle S_{0}(P,Q)=\cdots =S_{d-1}(P,Q)=0.}$

Every coefficient of the subresultant polynomials is defined as the determinant of a submatrix of the Sylvester matrix of P and Q. This implies that subresultants "specialize" well. More precisely, subresultants are defined for polynomials over any commutative ring R, and have the following property.

Let φ be a ring homomorphism of R into another commutative ring S. It extends to another homomorphism, denoted also φ between the polynomials rings over R and S. Then, if P and Q are univariate polynomials with coefficients in R such that

${\displaystyle \deg(P)=\deg(\varphi (P))}$

and

${\displaystyle \deg(Q)=\deg(\varphi (Q)),}$

then the subresultant polynomials and the principal subresultant coefficients of φ(P) and φ(Q) are the image by φ of those of P and Q.

The subresultants have two important properties which make them fundamental for the computation on computers of the GCD of two polynomials with integer coefficients. Firstly, their definition through determinants allows bounding, through Hadamard inequality, the size of the coefficients of the GCD. Secondly, this bound and the property of good specialization allow computing the GCD of two polynomials with integer coefficients through modular computation and Chinese remainder theorem (see below).

Let

${\displaystyle P=p_{0}+p_{1}X+\cdots +p_{m}X^{m},\quad Q=q_{0}+q_{1}X+\cdots +q_{n}X^{n}.}$

be two univariate polynomials with coefficients in a field K. Let us denote by ${\displaystyle {\mathcal {P}}_{i}}$ the K vector space of dimension i of polynomials of degree less than i. For non-negative integer i such that i ≤ m and i ≤ n, let

${\displaystyle \varphi _{i}:{\mathcal {P}}_{n-i}\times {\mathcal {P}}_{m-i}\rightarrow {\mathcal {P}}_{m+n-i}}$

be the linear map such that

${\displaystyle \varphi _{i}(A,B)=AP+BQ.}$

The resultant of P and Q is the determinant of the Sylvester matrix, which is the (square) matrix of ${\displaystyle \varphi _{0}}$ on the bases of the powers of X. Similarly, the i-subresultant polynomial is defined in term of determinants of submatrices of the matrix of ${\displaystyle \varphi _{i}.}$

Let us describe these matrices more precisely;

Let p_i = 0 for i < 0 or i > m, and q_i = 0 for i < 0 or i > n. The Sylvester matrix is the (m + n) × (m + n)-matrix such that the coefficient of the i-th row and the j-th column is p_m+j−i for j ≤ n and q_j−i for j > n:[2]

${\displaystyle S={\begin{pmatrix}p_{m}&0&\cdots &0&q_{n}&0&\cdots &0\\p_{m-1}&p_{m}&\cdots &0&q_{n-1}&q_{n}&\cdots &0\\p_{m-2}&p_{m-1}&\ddots &0&q_{n-2}&q_{n-1}&\ddots &0\\\vdots &\vdots &\ddots &p_{m}&\vdots &\vdots &\ddots &q_{n}\\\vdots &\vdots &\cdots &p_{m-1}&\vdots &\vdots &\cdots &q_{n-1}\\p_{0}&p_{1}&\cdots &\vdots &q_{0}&q_{1}&\cdots &\vdots \\0&p_{0}&\ddots &\vdots &0&q_{0}&\ddots &\vdots &\\\vdots &\vdots &\ddots &p_{1}&\vdots &\vdots &\ddots &q_{1}\\0&0&\cdots &p_{0}&0&0&\cdots &q_{0}\end{pmatrix}}.}$

The matrix T_i of ${\displaystyle \varphi _{i}}$ is the (m + n − i) × (m + n − 2i)-submatrix of S which is obtained by removing the last i rows of zeros in the submatrix of the columns 1 to n − i and n + 1 to m + n − i of S (that is removing i columns in each block and the i last rows of zeros). The principal subresultant coefficient s_i is the determinant of the m + n − 2i first rows of T_i.

Let V_i be the (m + n − 2i) × (m + n − i) matrix defined as follows. First we add (i + 1) columns of zeros to the right of the (m + n − 2i − 1) × (m + n − 2i − 1) identity matrix. Then we border the bottom of the resulting matrix by a row consisting in (m + n − i − 1) zeros followed by Xⁱ, Xⁱ⁻¹, ..., X, 1:

${\displaystyle V_{i}={\begin{pmatrix}1&0&\cdots &0&0&0&\cdots &0\\0&1&\cdots &0&0&0&\cdots &0\\\vdots &\vdots &\ddots &\vdots &\vdots &\ddots &\vdots &0\\0&0&\cdots &1&0&0&\cdots &0\\0&0&\cdots &0&X^{i}&X^{i-1}&\cdots &1\end{pmatrix}}.}$

With this notation, the i-th subresultant polynomial is the determinant of the matrix product V_iT_i. Its coefficient of degree j is the determinant of the square submatrix of T_i consisting in its m + n − 2i − 1 first rows and the (m + n − i − j)-th row.

It is not obvious that, as defined, the subresultants have the desired properties. Nevertheless, the proof is rather simple if the properties of linear algebra and those of polynomials are put together.

As defined, the columns of the matrix T_i are the vectors of the coefficients of some polynomials belonging to the image of ${\displaystyle \varphi _{i}}$. The definition of the i-th subresultant polynomial S_i shows that the vector of its coefficients is a linear combination of these column vectors, and thus that S_i belongs to the image of ${\displaystyle \varphi _{i}.}$

If the degree of the GCD is greater than i, then Bézout's identity shows that every non zero polynomial in the image of ${\displaystyle \varphi _{i}}$ has a degree larger than i. This implies that S_i=0.

If, on the other hand, the degree of the GCD is i, then Bézout's identity again allows proving that the multiples of the GCD that have a degree lower than m + n − i are in the image of ${\displaystyle \varphi _{i}}$. The vector space of these multiples has the dimension m + n − 2i and has a base of polynomials of pairwise different degrees, not smaller than i. This implies that the submatrix of the m + n − 2i first rows of the column echelon form of T_i is the identity matrix and thus that s_i is not 0. Thus S_i is a polynomial in the image of ${\displaystyle \varphi _{i}}$, which is a multiple of the GCD and has the same degree. It is thus a greatest common divisor.

Most root-finding algorithms behave badly with polynomials that have multiple roots. It is therefore useful to detect and remove them before calling a root-finding algorithm. A GCD computation allows detection of the existence of multiple roots, since the multiple roots of a polynomial are the roots of the GCD of the polynomial and its derivative.

After computing the GCD of the polynomial and its derivative, further GCD computations provide the complete square-free factorization of the polynomial, which is a factorization

${\displaystyle f=\prod _{i=1}^{\deg(f)}f_{i}^{i}}$

where, for each i, the polynomial f_i either is 1 if f does not have any root of multiplicity i or is a square-free polynomial (that is a polynomial without multiple root) whose roots are exactly the roots of multiplicity i of f (see Yun's algorithm).

Thus the square-free factorization reduces root-finding of a polynomial with multiple roots to root-finding of several square-free polynomials of lower degree. The square-free factorization is also the first step in most polynomial factorization algorithms.

The Sturm sequence of a polynomial with real coefficients is the sequence of the remainders provided by a variant of Euclid's algorithm applied to the polynomial and its derivative. For getting the Sturm sequence, one simply replaces the instruction

${\displaystyle r_{i+1}:=\operatorname {rem} (r_{i-1},r_{i})}$

of Euclid's algorithm by

${\displaystyle r_{i+1}:=-\operatorname {rem} (r_{i-1},r_{i}).}$

Let V(a) be the number of changes of signs in the sequence, when evaluated at a point a. Sturm's theorem asserts that V(a) − V(b) is the number of real roots of the polynomial in the interval [a, b]. Thus the Sturm sequence allows computing the number of real roots in a given interval. By subdividing the interval until every subinterval contains at most one root, this provides an algorithm that locates the real roots in intervals of arbitrary small length.

The content of a polynomial p ∈ R[X], denoted "cont(p)", is the GCD of its coefficients. A polynomial q ∈ F[X] may be written

${\displaystyle q={\frac {p}{c}}}$

where p ∈ R[X] and c ∈ R: it suffices to take for c a multiple of all denominators of the coefficients of q (for example their product) and p = cq. The content of q is defined as:

${\displaystyle \operatorname {cont} (q)={\frac {\operatorname {cont} (p)}{c}}.}$

In both cases, the content is defined up to the multiplication by a unit of R.

The primitive part of a polynomial in R[X] or F[X] is defined by

${\displaystyle \operatorname {primpart} (p)={\frac {p}{\operatorname {cont} (p)}}.}$

In both cases, it is a polynomial in R[X] that is primitive, which means that 1 is a GCD of its coefficients.

Thus every polynomial in R[X] or F[X] may be factorized as

${\displaystyle p=\operatorname {cont} (p)\,\operatorname {primpart} (p),}$

and this factorization is unique up to the multiplication of the content by a unit of R and of the primitive part by the inverse of this unit.

Gauss's lemma implies that the product of two primitive polynomials is primitive. It follows that

${\displaystyle \operatorname {primpart} (pq)=\operatorname {primpart} (p)\operatorname {primpart} (q)}$

and

${\displaystyle \operatorname {cont} (pq)=\operatorname {cont} (p)\operatorname {cont} (q).}$

The relations of the preceding section imply a strong relation between the GCD's in R[X] and in F[X]. To avoid ambiguities, the notation "gcd" will be indexed, in the following, by the ring in which the GCD is computed.

If q₁ and q₂ belong to F[X], then

${\displaystyle \operatorname {primpart} (\gcd _{F[X]}(q_{1},q_{2}))=\gcd _{R[X]}(\operatorname {primpart} (q_{1}),\operatorname {primpart} (q_{2})).}$

If p₁ and p₂ belong to R[X], then

${\displaystyle \gcd _{R[X]}(p_{1},p_{2})=\gcd _{R}(\operatorname {cont} (p_{1}),\operatorname {cont} (p_{2}))\gcd _{R[X]}(\operatorname {primpart} (p_{1}),\operatorname {primpart} (p_{2})),}$

and

${\displaystyle \gcd _{R[X]}(\operatorname {primpart} (p_{1}),\operatorname {primpart} (p_{2}))=\operatorname {primpart} (\gcd _{F[X]}(p_{1},p_{2})).}$

Thus the computation of polynomial GCD's is essentially the same problem over F[X] and over R[X].

For univariate polynomials over the rational numbers, one may think that Euclid's algorithm is a convenient method for computing the GCD. However, it involves simplifying a large number of fractions of integers, and the resulting algorithm is not efficient. For this reason, methods have been designed to modify Euclid's algorithm for working only with polynomials over the integers. They consist of replacing the Euclidean division, which introduces fractions, by a so-called pseudo-division, and replacing the remainder sequence of the Euclid's algorithm by so-called pseudo-remainder sequences (see below).

In the previous section we have seen that the GCD of polynomials in R[X] may be deduced from GCDs in R and in F[X]. A closer look on the proof shows that this allows us to prove the existence of GCDs in R[X], if they exist in R and in F[X]. In particular, if GCDs exist in R, and if X is reduced to one variable, this proves that GCDs exist in R[X] (Euclid's algorithm proves the existence of GCDs in F[X]).

A polynomial in n variables may be considered as a univariate polynomial over the ring of polynomials in (n − 1) variables. Thus a recursion on the number of variables shows that if GCDs exist and may be computed in R, then they exist and may be computed in every multivariate polynomial ring over R. In particular, if R is either the ring of the integers or a field, then GCDs exist in R[x₁,..., x_n], and what precedes provides an algorithm to compute them.

The proof that a polynomial ring over a unique factorization domain is also a unique factorization domain is similar, but it does not provide an algorithm, because there is no general algorithm to factor univariate polynomials over a field (there are examples of fields for which there does not exist any factorization algorithm for the univariate polynomials).

The simplest (to define) remainder sequence consists in taking always α = 1. In practice, it is not interesting, as the size of the coefficients grows exponentially with the degree of the input polynomials. This appears clearly on the example of the preceding section, for which the successive pseudo-remainders are

${\displaystyle -15\,X^{4}+3\,X^{2}-9,}$

${\displaystyle 15795\,X^{2}+30375\,X-59535,}$

${\displaystyle 1254542875143750\,X-1654608338437500,}$

${\displaystyle 12593338795500743100931141992187500.}$

The number of digits of the coefficients of the successive remainders is more than doubled at each iteration of the algorithm. This is typical behavior of the trivial pseudo-remainder sequences.

The primitive pseudo-remainder sequence consists in taking for α the content of the numerator. Thus all the r_i are primitive polynomials.

The primitive pseudo-remainder sequence is the pseudo-remainder sequence, which generates the smallest coefficients. However it requires to compute a number of GCD's in Z, and therefore is not sufficiently efficient to be used in practice, especially when Z is itself a polynomial ring.

With the same input as in the preceding sections, the successive remainders, after division by their content are

${\displaystyle -5\,X^{4}+X^{2}-3,}$

${\displaystyle 13\,X^{2}+25\,X-49,}$

${\displaystyle 4663\,X-6150,}$

${\displaystyle 1.}$

The small size of the coefficients hides the fact that a number of integers GCD and divisions by the GCD have been computed.

A subresultant sequence can be also computed with pseudo-remainders. The process consists in choosing α in such a way that every r_i is a subresultant polynomial. Surprisingly, the computation of α is very easy (see below). On the other hand, the proof of correctness of the algorithm is difficult, because it should take into account all the possibilities for the difference of degrees of two consecutive remainders.

The coefficients in the subresultant sequence are rarely much larger than those of the primitive pseudo-remainder sequence. As GCD computations in Z are not needed, the subresultant sequence with pseudo-remainders gives the most efficient computation.

With the same input as in the preceding sections, the successive remainders are

${\displaystyle 15\,X^{4}-3\,X^{2}+9,}$

${\displaystyle 65\,X^{2}+125\,X-245,}$

${\displaystyle 9326\,X-12300,}$

${\displaystyle 260708.}$

The coefficients have a reasonable size. They are obtained without any GCD computation, only exact divisions. This makes this algorithm more efficient than that of primitive pseudo-remainder sequences.

The algorithm computing the subresultant sequence with pseudo-remainders is given below. In this algorithm, the input (a, b) is a pair of polynomials in Z[X]. The r_i are the successive pseudo remainders in Z[X], the variables i and d_i are non negative integers, and the Greek letters denote elements in Z. The functions deg() and rem() denote the degree of a polynomial and the remainder of the Euclidean division. In the algorithm, this remainder is always in Z[X]. Finally the divisions denoted / are always exact and have their result either in Z[X] or in Z.

<span class="nowrap"><math>\begin{align}r_0:=a\\ r_1:=b \end{align}</math></span>
<span class="nowrap">'''''for''''' (<math>i:=1</math>; <math>r_i \neq 0</math>; <math>i:=i+1;</math>) '''''do'''''</span>
    <span class="nowrap"><math>\begin{align} d_{i}&:=\deg(r_{i-1})-\deg(r_{i})\\ \gamma_{i}&:=\operatorname{lc}(r_{i})
    \end{align}</math></span>
    <span class="nowrap">'''if''' <math>i=1</math> '''then'''</span>
        <span class="nowrap"><math>\begin{align} \beta_1&:=(-1)^{d_1+1}\\ \psi_1&:=-1 \end{align}</math></span>
    '''else'''
        <span class="nowrap"><math>\begin{align} \psi_{i}&:=\frac{(-\gamma_{i-1})^{d_{i-1}}}{\psi_{i-1}^{d_{i-1}-1}}\\ \beta_i&:=-\gamma_{i-1}\psi_i^{d_i} \end{align}</math></span>
    '''end if'''
    <span class="nowrap"><math>r_{i+1}:=\frac{\operatorname{rem}(\gamma_i^{d_i +1}r_{i-1},r_{i})}{\beta_i}</math></span>
'''''end do.'''''

Note: "lc" stands for the leading coefficient, the coefficient of the highest degree of the variable.

This algorithm computes not only the greatest common divisor (the last non zero r_i), but also all the subresultant polynomials: The remainder r_i is the (deg(r_i−1) − 1)-th subresultant polynomial. If deg(r_i) < deg(r_i−1) − 1, the deg(r_i)-th subresultant polynomial is lc(r_i)^{deg(r_i−1)−deg(r_i)−1}r_i. All the other subresultant polynomials are zero.

One may use pseudo-remainders for constructing sequences having the same properties as Sturm sequences. This requires to control the signs of the successive pseudo-remainders, in order to have the same signs as in the Sturm sequence. This may be done by defining a modified pseudo-remainder as follows.

If ${\displaystyle \deg(A)=a}$ and ${\displaystyle \deg(B)=b}$ and a ≥ b, the modified pseudo-remainder prem2(A, B) of the pseudo-division of A by B is

${\displaystyle \operatorname {prem2} (A,B)=-\operatorname {rem} (|\operatorname {lc} (B)|^{a-b+1}A,B),}$

where |lc(B)| is the absolute value of the leading coefficient of B (the coefficient of X^b).

For input polynomials with integer coefficients, this allows retrieval of Sturm sequences consisting of polynomials with integer coefficients. The subresultant pseudo-remainder sequence may be modified similarly, in which case the signs of the remainders coincide with those computed over the rationals.

Note that the algorithm for computing the subresultant pseudo-remainder sequence given above will compute wrong subresultant polynomials if one uses ${\displaystyle -\mathrm {prem2} (A,B)}$ instead of ${\displaystyle \operatorname {prem} (A,B)}$.