On an irreducibility criterion of Perron for multivariate polynomials

Transcription

1 Bull. Math. Soc. Sci. Math. Roumanie Tome 53(101) No. 3, 2010, On an irreducibility criterion of Perron for multivariate polynomials by Nicolae Ciprian Bonciocat Dedicated to the memory of Laurenţiu Panaitopol ( ) on the occasion of his 70th anniversary Abstract We combine a method of L. Panaitopol with some techniques for nonarchimedean absolute values to provide a new proof for an irreducibility criterion of Perron for multivariate polynomials over an arbitrary field. Key Words: Irreducibility, nonarchimedean absolute value Mathematics Subject Classification: Primary 11R09; Secondary 11C08. 1 Introduction A famous irreducibility criterion that requires no information on the canonical decomposition of the coefficients of an integer polynomial is the following result of Perron [5]. Theorem A (Perron) Let F(X) = X n +a n 1 X n 1 + +a 1 X +a 0 Z[X], with a 0 0. If a n 1 > 1 + a n a 1 + a 0, then F is irreducible in Z[X]. Using some techniques that require the study of sheets of a Riemann surface, Perron also stated in [5], in a slightly modified form, the following analogous irreducibility criterion for polynomials in two variables over an arbitrary field. Theorem B (Perron) Let K be a field, F(X,Y ) = a n (X)Y n + +a 1 (X)Y + a 0 K[X,Y ], with a 0,...,a n 1 K[X], a n K, a 0 a n 0. If deg a n 1 > max{deg a 0,deg a 1,...,deg a n 2 }, then F is irreducible over K(X). As an immediate consequence of Theorem B, one may formulate a similar irreducibility criterion for polynomials in r 3 variables X 1,X 2,...,X r over K.

2 214 N.C. Bonciocat For any polynomial f K[X 1,...,X r ] we denote by deg r f the degree of f as a polynomial in X r with coefficients in K[X 1,...,X r 1 ]. The next result follows from Theorem B by writing Y for X r, X for X r 1 and by replacing K with K(X 1,...,X r 2 ). Theorem C (Perron) Let K be a field, r 3, and let F = n i=0 a ix i r K[X 1,...,X r ] with a 0,...,a n 1 K[X 1,...,X r 1 ], a n K[X 1,...,X r 2 ] and a 0 a n 0. If deg r 1 a n 1 > max{deg r 1 a 0,deg r 1 a 1,...,deg r 1 a n 2 }, then F as a polynomial in X r is irreducible over K(X 1,...,X r 1 ). By a clever use of the triangle inequality, L. Panaitopol [4] obtained an elegant elementary proof of Theorem A, that makes no use of Rouché s Theorem. The aim of this note is to provide a new proof of Theorem B, based on ideas from [4] combined with the techniques used in [1], [2] and [3]. Before proceeding to the proof of Theorem B we note that the two conditions a n K and deg a n 1 > max{deg a 0,deg a 1,...,deg a n 2 } are best possible, in the sense that there exist polynomials in K[X,Y ] for which either a n / K and deg a n 1 > max{deg a 0,deg a 1,...,deg a n 2 }, or a n K and deg a n 1 = max{deg a 0,deg a 1,...,deg a n 2 }, and which are reducible over K[X]. To see this, one may first choose F 1 (X,Y ) = XY 2 + (X 2 + 1)Y + X. In this case deg a 1 > deg a 0, but a 2 / K, and F 1 is obviously reducible, since F 1 (X,Y ) = (XY + 1)(Y + X). For the second case one may choose F 2 (X,Y ) = Y 2 + (X + 1)Y + X. Here a 2 K, but deg a 1 = deg a 0, and F 2 is reducible too, since F 2 (X,Y ) = (Y + X)(Y + 1). 2 Proof of Theorem B. We will give a proof based on the study of the location of the roots of F, regarded as a polynomial in Y with coefficients in K[X]. We first introduce a nonarchimedean absolute value on K(X), as follows. We fix an arbitrary real number ρ > 1, and for any polynomial u(x) K[X] we define u(x) by the equality u(x) = ρ deg u(x). We then extend the absolute value to K(X) by multiplicativity. Thus for any w(x) K(X), w(x) = u(x) v(x), with u(x),v(x) K[X], v(x) 0, we let w(x) = u(x) v(x). Let us note that for any non-zero element u of K[X] one has u 1. Let now K(X) be a fixed algebraic closure of K(X), and let us fix an extension of our absolute value to K(X), which we will also denote by. Using our absolute value, the condition deg a n 1 > max{deg a 0,deg a 1,...,deg a n 2 }

3 On an irreducibility criterion of Perron 215 reads We also have a n 1 > max{ a 0, a 1,..., a n 2 }. (1) a 0 a n = 1. (2) We consider now the factorisation of the polynomial F(X,Y ) over K(X), say F(X,Y ) = a n (X)(Y θ 1 ) (Y θ n ), with θ 1,...,θ n K(X). Since a 0 0 we must have θ i 0, i = 1,...,n. We will prove now that conditions (1) and (2) force F to have a single root θ with θ > 1, and all the other roots with θ < 1. To see this, we will first prove that F has no roots θ with θ = 1. Indeed, if F would have a root θ with θ = 1, then a n 1 θ n 1 = a n θ n + a n 2 θ n a 1 θ + a 0 and hence a n 1 = a n 1 θ n 1 = a n θ n + a n 2 θ n a 1 θ + a 0 max{ a n θ n, a n 2 θ n 2,..., a 1 θ, a 0 } = max{ a n θ n, a n 2 θ n 2,..., a 1 θ, a 0 } = max{ a n, a n 2,..., a 1, a 0 }, which cannot hold, according to (1) and (2). On the other hand θ 1 θ n = a 0 / a n 1, so F must have at least one root θ with θ > 1, say θ = θ 1. Therefore we may write F as F(X,Y ) = (Y θ 1 ) G(X,Y ), with G(X,Y ) = a n (Y θ 2 ) (Y θ n ) = = b n 1 Y n 1 + b n 2 Y n b 1 Y + b 0 K(X)[Y ]. Equating the coefficients in the equality a n Y n + + a 1 Y + a 0 = (Y θ 1 )(b n 1 Y n b 1 Y + b 0 ), we deduce that a 0 = θ 1 b 0 a i = b i 1 θ 1 b i for i = 1,2,...,n 1 a n = b n 1. (3) Note that by (1) and (2) we have a n 1 > max{ a 0, a 1,..., a n 2, a n }. In what follows, we will use a stronger inequality, which is the key factor for the remaining part of the proof. More precisely, since deg a n 1 > max{deg a 0,deg a 1,...,deg a n 2 }

4 216 N.C. Bonciocat and deg a 0 deg a n, we observe that for sufficiently large ρ we actually have that is ρ deg an 1 > ρ deg a0 + ρ deg a1 + + ρ deg an 2 + ρ deg an, a n 1 > a 0 + a a n 2 + a n. (4) Using (3), (4) and the fact that our absolute value also satisfies the triangle inequality, we obtain b n 2 + θ 1 b n 2 θ 1 b n 1 = a n 1 > a 0 + a a n 2 + a n = θ 1 b 0 + b 0 θ 1 b 1 + b 1 θ 1 b b n 3 θ 1 b n θ 1 b 0 + ( θ 1 b 1 b 0 ) + ( θ 1 b 2 b 1 ) + + +( θ 1 b n 2 b n 3 ) + 1 = θ 1 ( b 0 + b b n 2 ) ( b 0 + b b n 3 ) + 1 = ( θ 1 1) ( b 0 + b b n 2 ) + b n 2 + 1, which yields θ 1 1 > ( θ 1 1) ( b 0 + b b n 2 ). After division by θ 1 1, we obtain b 0 + b b n 2 < 1. (5) We will prove now that (5) forces the roots θ 2,...,θ n of G to have all absolute values strictly less than 1. Indeed, if we assume that G has a root θ with θ 1, then θ n 1 = b n 1 θ n 1 = b n 2 θ n b 1 θ + b 0 b n 2 θ n b 1 θ + b 0 θ n 1 ( b n b 1 + b 0 ), and hence b n b 1 + b 0 1, which contradicts (5). Therefore θ i < 1 for i = 2,...,n. We can prove now that F is irreducible over K(X). Let us assume by the contrary that F decomposes as F(X,Y ) = F 1 (X,Y ) F 2 (X,Y ), with F 1, F 2 K(X)[Y ], deg Y F 1 = t 1 and deg Y F 2 = s 1. By the celebrated lemma of Gauss we may in fact assume that F 1, F 2 K[X,Y ]. Without loss of generality we may further assume that θ 1 is a root of F 2, which implies that all the roots of F 1 have absolute values strictly less than 1. On the other hand, if we write F 1 as F 1 (X,Y ) = c 0 + c 1 Y + + c t Y t, say, with c i K[X], i = 0,...,t, then we must have c 0 a 0 and c t a n. This shows that c t K, that is c t = 1, so the absolute value of the product of the roots of F 1 is c 0 / c t = c 0 1, which is a contradiction. This completes the proof of the theorem. Acknowledgements. This work was supported by CNCSIS-UEFISCSU, project PNII-IDEI 443, code 1190/2008. The author expresses his gratitude to Professor Mihai Cipu for useful discussions on various techniques for testing irreducibility.

5 On an irreducibility criterion of Perron 217 References [1] A.I. Bonciocat, N.C. Bonciocat, A Capelli type theorem for multiplicative convolutions of polynomials, Math. Nachr. 281 (2008), no. 9, [2] A.I. Bonciocat, N.C. Bonciocat, Some classes of irreducible polynomials, Acta Arith. 123 (2006) no. 4, [3] M. Cavachi, M. Vajaitu, A. Zaharescu, An irreducibility criterion for polynomials in several variables, Acta Math. Univ. Ostrav. 12 (2004), no. 1, [4] L. Panaitopol, Criteriul lui Perron de ireductibilitate a polinoamelor cu coeficienti intregi, Gaz. Mat. vol. XCVIII no. 10 (1993), [5] O. Perron, Neue kriterien für die irreduzibilität algebraischer gleichungen, J. reine angew. Math. 132 (1907), Received: Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest , Romania, Nicolae.Bonciocat@imar.ro