Download Polynomials in Rings: Operations, Euclidean Division, and Rational Fractions and more Lecture notes Computer Science in PDF only on Docsity!
Chapitre 4
Concept of polynomial with an
indeterminate coe¢ cient in a ring
Let K be a Öeld (K =Q or K =R or K =C).
DÈÖnition 4.1 A polynomial with coe¢ cients in K is an expression of the form :
P (X) = anXn^ + Xn 1 Xn ^1 + ::: + a 1 X + a 0 ;
where n 2 N and faigi=1; 2 ;:::;n 2 K is called coe¢ cients of the polynomial.
Remarque 4.1 The set of polynomials on the Öeld K is denoted K [X].
DÈÖnition 4.2 Let 0 6 = P (X) 2 K [X]. We call the degree of P (X) the largest integer n such that an 6 = 0, we denote it deg(P ), and the element adeg(P ) is called the dominant coe¢ cient of P.
Remarque 4.2 The degree of the zero polynomial adeg(P =0) = 1 by convention.
DÈÖnition 4.3 Let 0 6 = P (X) 2 K [X]. We say that P is unitary (normalized) if and only adeg(P ) = 1.
DÈÖnition 4.4 A one-indeterminate polynomial with coe¢ cients in K [X] whose coe¢ - cients are all zero from a certain rank.
4.1 Operations on K [X]
Consider the two polynomials P (X) = anXn^ + Xn 1 Xn ^1 + ::: + a 1 X + a 0 = Pn i=0^ aiX
i
and Q (X) = bnXn^ + bn 1 Xn ^1 + ::: + b 1 X + b 0 = Pn l=0^ bj^ X
j (^). We deÖne addition, product of polynomials and multiplication by a scalar 2 K as follows : ñ (P + Q) (X) = Pn i=0^ ciX
i (^) where ci = ai + bi.
ñ (P Q) (X) = P^2 n k= vkXk^ where vk = Pn k= akbn k. ñ (P ) (X) = Pn i=0^ ciX
i. ñ P (X) = Q (X) , 8i 2 N; ai = bi.
DÈÖnition 4.5 The addition and multiplication deÖned above deÖne internal composi- tion laws on the set of polynomials with an indeterminate coe¢ cient in K.
DÈÖnition 4.6 The set of polynomials with an indeterminate coe¢ cient in K provided with addition and multiplication deÖned a commutative ring structure which we denote K [X].
4.2 Arithmetic of polynomials
DÈÖnition 4.7 Let P and Q be two polynomials of K [X], We say that the polynomial P is divisible by the polynomial Q if there exists a polynomial A such that P = QA and we write QnP and we say that P is multiple of Q (where Q is a divisor of P ).
Proposition 4.1 Let P; Q; R 2 K [X] we have : ñ P nQ and QnP ) 9 2 K : P = Q. ñ P nQ and QnR ) P nR. ñ P nQ and P nR ) P n (Q + R) where ; 2 K. ñ P nP; 1 nP and P n 0.
DÈÖnition 4.11 Let m 2 N. We say that is a root of multiplicity m of P if (X )m^ nP while (X )m+1^ does not divide P.
Remarque 4.3 When m = 1, is called a simple root. If m > 1 , is called a root of order m.
Proposition 4.4 Let P 2 K [X] and 2 K such that is a root of order m 2 N^ f 1 g, the following assertions are equivalent ñ 9 Q 2 K [X] : P = (X )m^ Q with Q ( ) 6 = 0. ñ P ( ) = 0; P 0 ( ) = 0; :::; P (m 1)^ ( ) = 0 and P (m)^ ( ) 6 = 0 where P (i)^ is the deriva- tive of order i of P.
4.3 Concept of rational fraction with an indetermi-
nate
We denote by K [X]^ the set of non-zero polynomials i.e. K [X]^ = K [X] n (^0) K[X] and we consider on K [X] K [X]^ the equivalence relation R deÖned as follows :
8 (P; Q) ; (A; B) 2 K [X] K [X]^ ; (P; Q) R (A; B) , P B = Q A:
DÈÖnition 4.12 We call the rational fraction on K the equivalence class of (P; Q) 2 K [X] K [X]^ and is denoted P=Q or PQ i.e.
P Q =^ f(A; B)^2 K^ [X]^ ^ K^ [X]
(^) : P B = Q Ag :
Remarque 4.4 We note by K (X) = K [X]K [X]^ =R = ^ AB : (A; B) 2 K [X] K [X]^.
DÈÖnition 4.13 We call irreducible form of a non-zero rational fraction P of K (X) any couple (A; B) 2 K [X]^ K [X]^ : gcd (A; B) = 1K[X].
4.3.1 Operations on K (X)
The set K (X) endowed with the following two laws of internal composition 8 PQ ; AB 2 K (X) ; PQ + AB = P^ BQ+QB A and PQ AB = PQ^ AB has a commutative Öeld structure.
Canonical injection of K [X] into K (X)
Consider the map J : K [X]! K (X) which associates with each polynomial P of K [X] the rational fraction (^1) KP[X]. We can easily prove that J is injective. So we can identify the elements of K [X] with the elements of K (X) and we have K [X] K (X).
Remarque 4.5 J is called canonical injection.
Roots and poles of a rational fraction
DÈÖnition 4.14 Consider the irreducible rational fraction F = PQ 2 K (X). ñ The roots of F are the zeros of P in K [X]. ñ The order of multiplicity of the root of F is the same when considering it as the root of P in K [X]. ñ The poles of F are the zeros of Q in K [X]. ñ The order of multiplicity of the pole of F is the same when considering it as the root of Q in K [X].
DÈÖnition 4.15 Consider the rational fraction F = PQ 2 K (X). We call an associated function to F any function Fe : K! K deÖned for all x di§erent from the poles of F i.e. the application Fe (x) = PQe e^ ((xx)).
4.3.2 Decomposition of a rational fraction
Consider the irreducible rational fraction F = PQ 2 K (X). By means of the Euclidean division in K [X] we have the existence and the uniqueness of two polynomials D and R
Remarque 4.9 In the case where Q = X the partial sum is called the polar part relative to.
Remarque 4.10 In the case where Q = X , we speak of the decomposition into simple elements of the Örst kind. On the other hand if Q = aX^2 + bX + c we speak of decomposition into simple elements of the second kind.
