On the factorization of polynomials over discrete valuation domains

Transcription

1 DOI: /auom An. Şt. Univ. Oviius Constanţa Vol. 22(1),2014, On the factorization of polynomials over iscrete valuation omains Doru Ştefănescu Abstract We stuy some factorization properties for univariate polynomials with coefficients in a iscrete valuation omain (A, v). We use some properties of the Newton inex of a polynomial F (X) = i=0 aix i A[X] to euce conitions on v(a i) that allow us to fin some information on the egree of the factors of F. 1 Introuction One of the olest irreucibility criterion for univariate polynomials with coefficients in a valuation omain was given by G. Dumas [10] as a valuation approach to Schönemann-Eisenstein s criterion for polynomials with integer coefficients ([21] an [11]). Theorem 1.1. Let F (X) = i=0 a ix i be a polynomial over a iscrete valuation omain A, with value fiel (K, v). If the following conitions are fulfille i) v(a 0 ) = 0, ii) v(a ) < v(ai) i for all i {1, 2,..., 1}, iii) (v(a ), ) = 1, then the polynomial F (X) is irreucible in K[X]. Key Wors: irreucible polynomials, valuation, Newton polygon 2010 Mathematics Subject Classification: Primary 11R09; Seconary 11C08. Receive: November, Accepte: November,

2 274 Doru Ştefănescu There are many recent results that provie irreucibility conitions for various classes of polynomials by using techniques coming from valuation theory (see for instance [23], [24], [2], [3], [4], [8], [5] an [9]), or Newton polygon metho (see for instance [12], [13], [14], [15], [16], [17], [18], [1], [6], [7], [22] an [25]). In this paper we will consier univariate polynomials F (X) = i=0 a ix i with coefficients in a iscrete valuation omain (A, v), an we will stuy some of their factorization properties by using information on the quotients v(a 0) v(a i) i. The results obtaine for such polynomials are relate to the irreucibility criteria of Schönemann, Eisenstein, Dumas an their generalizations, but also to the irreucibility of generalize ifference polynomials (see for instance [20] an [19]). Throughout this paper we will suppose that v(a 0 ) = 0. Our results will require the use of the Newton inex of the polynomial F, which is efine by e(f ) = max 1 i v(a 0 ) v(a i ). i We note here that the value of the Newton inex in Dumas theorem an its extensions is attaine for the couple (, v(a )), so the Newton inex of the polynomial F is in this case v(a )/. In this paper we will also consier the case in which the maximum in the efinition of e(f ) may be attaine for an inex i. 2 Main results With the notations in previous section, we have Proposition 2.1. If F 1 F 2 A[X] \ A then e(f 1 F 2 ) = max (e(f 1 ), e(f 2 )). Proof: We remin that one may associate to the polynomial F its Newton polygon N(F ) efine as the lower convex hull of the set of points {(0, v(a )), (1, v(a 1 )),..., (, v(a 0 ))} By the celebrate theorem of Dumas [10], we know that if F = F 1 F 2 is a nontrivial factorization of F in A[X], then the eges of the Newton polygon of F can be constructe through translates of those of the Newton polygons N(F 1 ) an N(F 2 ), using exactly one translate for each ege, in such a way as to form a polygonal path with the slopes of the eges increasing.

3 ON THE FACTORIZATION OF POLYNOMIALS OVER VALUATION DOMAINS275 Using this result, it will be sufficient to observe that the quotient v(a0) v(ai) i is the slope of the line joining the points (, v(a 0 )) an ( i, v(a i )). In the following result we will suppose that the Newton inex of F is attaine for an inex i {1,..., }, fact that will not necessarily imply the irreucibility of F, but will allow us in case F is reucible to obtain some information on the egree of one of its factors. Theorem 2.2. Let (A, v) be a iscrete valuation omain, an let F (X) = a 0 X + a 1 X a 1 X + a A[X]. Suppose that v(a 0 ) = 0 an that there exists an inex s {1, 2,..., } such that the following conitions are satisfie: (a) v(a s) < v(a i) for i {1, 2,..., }, i s; s i (b) sv(a ) v(a s ) = 1. (c) (s, v(a s )) = 1; Then the polynomial F is either irreucible in A[X], or has a factor whose egree is a multiple of s. Proof: Let us assume that there exists a nontrivial factorization F = F 1 F 2 of the polynomial F in A[X], an let us enote an We also put = eg F, 1 = eg(f 1 ) 1, 2 = eg(f 2 ) 1, m = v(a ), a = v(a s ). m 1 = v(f 1 (0)) an m 2 = v(f 2 (0)). With these notations, conition (b) reas sm a = 1. (1) Now, since conition (a) shows that e(f ) = v(as) s, an by Proposition 2.1 we have e(f ) = max{e(f 1 ), e(f 2 )}, we first euce that so we must have a s = v(a s) s = e(f ) e(f 1 ) v(f 1(0)) 1 = m 1 1, a 1 sm 1. (2)

4 276 Doru Ştefănescu On the other han, since = eg(f 1 F 2 ) = eg(f 1) + eg(f 2 ) = an m = v(f 1 (0)F 2 (0)) = v(f 1 (0)) + v(f 2 (0)) = m 1 + m 2, we see by (1) an (2) that Now, since sm 2 a 2 sm a = 1. a s = e(f ) e(f 2) m 2 2, we euce that a 2 sm 2, so we must have 0 sm 2 a 2 1. (3) Next, since sm 2 a 2 is an integer, (3) shows that it can only take the value 0 or 1, so we istinguish two cases: Case 1: sm 2 a 2 = 0. Here, since conition (b) implies in particular the fact that a an s are coprime, we see that 2 must be ivisible by s. Case 2: sm 2 a 2 = 1. In this case we have s(m m 1 ) a( 1 ) = 1, which in view of (1) shows that sm 1 = a 1, an since a an s are coprime, we see now that 1 must be ivisible by s. Therefore, if the polynomial F is reucible, the egree of one of its factors must be a multiple of s. With the notations in Theorem 2.2, one has the following result. Corollary 2.3. If 4 an s > /2, then the polynomial F is either irreucible, or has a ivisor of egree s. Proof: If F woul have a factor of egree ks, with k 2, then we woul obtain a contraiction. > ks > k 2,

5 ON THE FACTORIZATION OF POLYNOMIALS OVER VALUATION DOMAINS277 3 Examples 1) Let F (X) = X + p (p 1)X 2 + p 2 X + p 1 Z[X], with 3 an p a prime number, an let us consier the usual p aic value on Z, enote by v. Since an v(a 1 ) 1 = 2 1 < 2 = v(a 2) 2 v(a 1 ) 1 = 2 1 < 1 = v(a ), we may take s = 1, an since sv(a ) v(a s ) = ( 1) 2 ( 2) = 1, we conclue by Theorem 2.2 that F is either irreucible, or has a factor of egree 1, an hence also a linear factor. On the other han, one may easily check that F has no integer solutions, an hence is an irreucible polynomial. 2) Let F (X, Y ) = Y + q(x)y + r(x) Z[X, Y ], where q, r Z[X] with eg(q) = eg(r) = 1. Using now the iscrete valuation on Z[X] given by v(h) = eg(h) for h Z[X], we see that v(q) 1 = 1 1 < 1 = v(r), so with the notation in Theorem 2.2 we have s = 1. On the other han, using the same notation we observe that sv(a ) v(a s ) = ( 1)v(r) v(q) = 1. It follows that F is either irreucible in Z[X, Y ], or has a linear factor with respect to Y. 3) Let K be a fiel of characteristic zero, 4 an integer, an let F (X, Y ) = Y + (X 2 + 1)Y 2 + (X + X + 1)Y + X +1 + X K[X, Y ]. We represent the polynomial F as F (X, Y ) = Y + a 2 (X)Y 2 + a 1 (X)Y + a (X) with a 2 (X) = X 2 +1, a 1 (X) = X +X +1 an a (X) = X +1 +X Using now the iscrete valuation on K[X] given by v(h) = eg(h) for h K[X], we observe that v(a 2 ) 2 = 1, v(a 1 ) 1 = 1 an v(a ) = + 1. < v(a 2) an v(a 1) 1 < v(a ) Therefore v(a 1) 1 2, so we may take s = 1, an since sv(a ) v(a s ) = 1, we conclue by Theorem 2.2 that F is either

6 278 Doru Ştefănescu irreucible in K[X, Y ], or has a factor whose egree with respect to Y is a multiple of 1, that is F is either irreucible, or has a linear factor in Y. Acknowlegement. The publication of this paper was supporte by the grants PN-II-ID-WE an PN-II-ID-WE References [1] A. Bishnoi, S. K. Khanuja, K. Suesh, Some extensions an applications of the Eisenstein irreucibility criterion, Developments in Mathematics 18 (2010), [2] A.I. Bonciocat, N.C. Bonciocat, Some classes of irreucible polynomials, Acta Arith. 123 (2006), no. 4, [3] A.I. Bonciocat, N.C. Bonciocat, A Capelli type theorem for multiplicative convolutions of polynomials, Math. Nachr. 281 (2008), no. 9, [4] N.C. Bonciocat, On an irreucibility criterion of Perron for multivariate polynomials, Bull. Math. Soc. Sci. Math. Roumanie 53 (101) No. 3 (2010), [5] N.C. Bonciocat, From prime numbers to irreucible multivariate polynomials, An. Ştiint. Univ. Oviius Constanţa Ser. Mat. 19 (2011) no. 2, pag [6] N.C. Bonciocat, Schönemann-Eisenstein-Dumas-type irreucibility conitions that use arbitrarily many prime numbers, arxiv: v1. [7] N.C. Bonciocat, Y. Bugeau, M. Cipu, M. Mignotte, Irreucibility criteria for sums of two relatively prime polynomials, Int. J. Number Theory 9 (6) (2013), [8] N.C. Bonciocat, A. Zaharescu, Irreucible multivariate polynomials obtaine from polynomials in fewer variables, J. Pure Appl. Algebra 212 (2008), no. 10, [9] N.C. Bonciocat, A. Zaharescu, Irreucible multivariate polynomials obtaine from polynomials in fewer variables, II, Proc. Inian Aca. Sci. Math. Sci. 121 (2011) no. 2, [10] G. Dumas, Sur quelques cas irréucibilité es polynômes à coefficients rationnels, Journal e Math. Puers et Appl., 12 (1906),

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8 280 Doru Ştefănescu [25] S.H. Weintraub, A mil generalization of Eisenstein s criterion, Proc. Amer. Math. Soc. 141 (2013), no. 4, Doru Ştefănescu Department of Theoretical Physics an Mathematics, University of Bucharest Bul M. Kogălniceanu 36 46, Bucharest, Romania stef@rms.unibuc.ro