DECOMPOSITION OF POLYNOMIALS AND APPROXIMATE ROOTS

Description

Niveau: Supérieur, Doctorat, Bac+8
DECOMPOSITION OF POLYNOMIALS AND APPROXIMATE ROOTS ARNAUD BODIN Abstract. We state a kind of Euclidian division theorem: given a polynomial P (x) and a divisor d of the degree of P , there exist polynomials h(x), Q(x), R(x) such that P (x) = h ? Q(x) + R(x), with deg h = d. Under some conditions h,Q,R are unique, and Q is the approximate d-root of P . Moreover we give an algorithm to compute such a decomposition. We apply these results to decide whether a polynomial in one or several variables is decomposable or not. 1. Introduction Let A be an integral domain (i.e. a unitary commutative ring without zero divisors). Our main result is: Theorem 1. Let P ? A[x] be a monic polynomial. Let d > 2 such that d is a divisor of degP and d is invertible in A. There exist h,Q,R ? A[x] such that P (x) = h ?Q(x) + R(x) with the conditions that (i) h,Q are monic; (ii) deg h = d, coeff(h, xd?1) = 0, degR < degP ? degPd ; (iii) R(x) = ∑ i rix i with (degQ|i ? ri = 0).

• moh who


• called abhyankar-moh


• unique monic polynomial


• thd step


• coefficients before


• monic polynomial

Sujets

• Suzuki
• Polynomial

Informations

• Publié par: mijec
• Nombre de lectures: 12
• Langue: English

Extrait

DECOMPOSITION OF POLYNOMIALS AND APPROXIMATE ROOTS

ARNAUD BODIN

Abstract.givenWe state a kind of Euclidian division theorem: a polynomialP(x) and a divisordof the degree ofP, there exist polynomialsh(x), Q(x), R(x) such thatP(x) =h◦Q(x) +R(x), with degh=d. Under some conditionsh, Q, Rare unique, andQ is the approximated-root ofPwe give an algorithm to. Moreover compute such a decomposition. We apply these results to decide whether a polynomial in one or several variables is decomposable or not.

1.Introduction LetAbe an integral domain (i.e. a unitary commutative ring without zero divisors). Our main result is: Theorem 1.LetP∈A[x]Letbe a monic polynomial. d>2such that dis a divisor ofdegPanddis invertible inA. There existh, Q, R∈ A[x]such that P(x) =h◦Q(x) +R(x) with the conditions that (i)h, Qare monic; d− 1 degP (ii) degh=d,coeff(h, x) = 0,degR <degP−; d P i (iii)R(x) =r i xwith(degQ|i⇒r i = 0). i Moreover suchh, Q, Rare unique. The previous theorem has a formulation similar to the Euclidian division; but hereQis not given (only its degree is fixed); there is a naturalQ(that we will compute, see Corollary 2) associated toP andd. Notice also that the decompositionP(x) =h◦Q(x) +R(x) is nottheQ-adic decomposition, since the coefficients before the powers i Q(x) belong toAand not toA[x].

Date: October 16, 2009. 2000Mathematics Subject Classification.13B25. Key words and phrases.Decomposable and indecomposable polynomials in one or several variables. 1