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Polynomial greatest common divisor - Wikipedia, the free encyclopedia. In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common divisor of two integers.
In the important case of univariate polynomials over a field the polynomial GCD may be computed, like for the integer GCD, by Euclid's algorithm using long division. The polynomial GCD is defined only up to the multiplication by an invertible constant.
The similarity between the integer GCD and the polynomial GCD allows us to extend to univariate polynomials all the properties that may be deduced from Euclid's algorithm and Euclidean division. Moreover, the polynomial GCD has specific properties that make it a fundamental notion in various areas of algebra. Typically, the roots of the GCD of two polynomials are the common roots of the two polynomials, and this allows us to get information on the roots without computing them. For example, the multiple roots of a polynomial are the roots of the GCD of the polynomial and its derivative, and further GCD computations allow us to compute the square- free factorization of the polynomial, which provides polynomials whose roots are the roots of a given multiplicity.
The greatest common divisor may be defined and exists, more generally, for multivariate polynomials over a field or the ring of integers, and also over a unique factorization domain. There exist algorithms to compute them as soon as one has a GCD algorithm in the ring of coefficients. These algorithms proceed by a recursion on the number of variables to reduce the problem to a variant of Euclid's algorithm. They are a fundamental tool in computer algebra, because computer algebra systems use them systematically to simplify fractions. Conversely, most of the modern theory of polynomial GCD has been developed to satisfy the need of efficiency of computer algebra systems. General definition. A greatest common divisor of p and q is a polynomial d that divides p and q and such that every common divisor of p and q also divides d.
Every pair of polynomials (not both zero) has a GCD if and only if F is a unique factorization domain. If F is a field and p and q are not both zero, for d to be a greatest common divisor it is sufficient that it divides both p and q and it has the greatest degree among the polynomials having this property. If p = q = 0, the GCD is 0.
Setting a+bi makes sure that the program displays imaginary numbers as well. Factorization of Polynomials and Real Analytic Functions Radoslaw (Radek). This can certainly be done (for large enough r). Best Answer: Factor Any Polynomial is one of the best ones available to use.
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However, some authors consider that it is not defined in this case. The greatest common divisor of p and q is usually denoted . With this convention, the GCD of two integers is also the greatest (for the usual ordering) common divisor. However, since there is no natural total order for polynomials over an integral domain, one cannot proceed in the same way here. For univariate polynomials over a field, one can additionally require the GCD to be monic (i.
Therefore, equalities like d = gcd(p, q) or gcd(p, q) = gcd(r, s) are usual abuses of notation which should be read . In particular, gcd(p, q) = 1 means that the invertible constants are the only common divisors, and thus that p and q are coprime. Properties. This property is at the basis of the proof of Euclid's algorithm. For any invertible element k of the ring of the coefficients, gcd(p,q)=gcd(p,kq). It may be computed recursively from GCD's of two polynomials by the identities: gcd(p,q,r)=gcd(p,gcd(q,r)).
Two of them are: Factorization of polynomials, in which one finds the factors of each expression, then selects the set of common factors held by all from within each set of factors. This method may be useful only in simple cases, as factoring is usually more difficult than computing the greatest common divisor. The Euclidean algorithm, which can be used to find the GCD of two polynomials in the same manner as for two numbers. Factoring. 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 x.
The Euclidean algorithm is a method which works for any pair of polynomials. It makes repeated use of polynomial long division or synthetic 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, assume we wish to find the gcd of two polynomials a(x) and b(x), where we can supposedeg. Notice that a polynomial g(x) divides a(x) and b(x) if and only if it divides b(x) and r. We deducegcd(a(x),b(x))=gcd(b(x),r. At each stage we havedeg. If the coefficients are floating point numbers, known only approximately, then one uses completely different techniques, usually based on SVD.
This induces a new difficulty: For all these fields except the finite ones, the coefficients are fractions. If the fractions are not simplified during the computation, the size of the coefficients grows exponentially during the computation, which makes it impossible except for very small degrees. On the other hand, it is highly time consuming to simplify the fractions immediately.
Therefore, two different alternative methods have been introduced (see below): Pseudo- remainder sequences, especially subresultant sequences. Modular GCD algorithm using modular arithmetic. Univariate polynomials with coefficients in a field. Firstly, it is the most elementary case and therefore appear in most first courses in algebra. Secondly, it is very similar to the case of the integers, and this analogy is the source of the notion of Euclidean domain. A third reason is that the theory and the algorithms for the multivariate case and for coefficients in a unique factorization domain are strongly based on this particular case.
Last but not least, polynomial GCD algorithms and derived algorithms allow one to get useful information on the roots of a polynomial, without computing them. Euclidean division. Its existence is based on the following theorem: Given two univariate polynomials a and b . 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 . Thus the proof of the validity of this algorithm also proves the validity of Euclidean division. Euclid's algorithm.
The GCD is the last non zero remainder. Euclid's algorithm may be formalized in the recursive programming style as: gcd(a,b): =ifb=0thenaelsegcd(b,rem(a,b)). Thus after, at most, deg(b) steps, one get a null remainder, say rk. 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 (ri, ri + 1) have the same set of common divisors. The common divisors of a and b are thus the common divisors of rk . This not only proves that Euclid's algorithm computes GCDs, but also proves that GCDs exist. B. 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, then there are two polynomials u and v such thatau+bv=g(Be. It is therefore called extended GCD algorithm.
Another difference with Euclid's algorithm is that it also uses the quotient, denoted . This algorithm works as follows.
Extended GCD algorithm. Input: a, b, univariate polynomials. Output: g,the GCD of a and bu, v, such that. 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: a+Lb=a+K. 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). Subresultants. In fact 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 to characterize generically the GCD of two polynomials, and the resultant is the 0- th subresultant polynomial. They have the property that the GCD of P and Q has a degree d if and only ifs. P,Q)=. This implies that subresultants .
More precisely, subresultants are defined for polynomials over any commutative ring R, and have the following property. Let . It extends to another homomorphism, denoted also . Then, if P and Q are univariate polynomials with coefficients in R such thatdeg. Firstly, their definition through determinants allows to bound, through Hadamard inequality, the size of the coefficients of the GCD.
Secondly, this bound and the property of good specialization allow to compute the GCD of two polynomials with integer coefficients through modular computation and Chinese remainder theorem (see below). Technical definition. For non negative integer i such that i. Similarly, the i- subresultant polynomial is defined in term of determinants of submatrices of the matrix of . The Sylvester matrix is the (m + n) . The principal subresultant coefficientsi is the determinant of the m + n - 2i first rows of Ti. Let Vi be the (m + n .