An irreducibility criterion for power series
aa r X i v : . [ m a t h . C V ] M a y AN IRREDUCIBILITY CRITERION FOR POWER SERIES
GUILLAUME ROND AND BERND SCHOBER
Abstract.
We prove an irreducibility criterion for polynomials with power seriescoefficients generalizing previous results given in [GBGP] and [ACLM1]. Introduction
The aim of this note is to provide a natural approach to an irreducibility criterion forpolynomials with power series coefficients (see Theorem 2.3). The first version of thecriterion has been given in [GBGP] and then has been generalized in [ACLM1]. In thisnote we give a more natural and elementary proof of a general version of this criterion.In particular, our statement holds over any field while the previous ones were only provenfor algebraically closed fields of characteristic zero. Moreover, the only hypothesis thatwe need is that the projection of the Newton polyhedron has exactly one vertex while theprevious known versions were involving additional technical conditions.Let us recall that the proof given in [GBGP] uses toric geometry and Zariski Main Theoremwhile the one provided in [ACLM1] is based on a generalization of the Newton’s methodfor plane curves. Our proof is essentially based on the following well known version ofHensel’s Lemma:
Proposition 1.1 (Hensel’s Lemma) . Let ( R, m ) be a Henselian local ring. A monicpolynomial P ( Z ) ∈ R [ Z ] , that is the product of two monic coprime polynomials modulo m R [ Z ] , is in fact the product of two coprime monic polynomials. We begin by giving some definitions and our main result (Theorem 2.3). In a second partwe give an example showing that our main result cannot be extended in a more generalsetting.Finally, let us mention that this irreducibility criterion is very useful in the study ofquasi-ordinary hypersurfaces (see [ACLM2] or [MS]).2.
An irreducibility criterion
We denote by k [[ x ]] the ring of formal power series in n variables x := ( x , . . . , x n ) over afield k . For any vector β ∈ Z n we set x β := x β · · · x β n n and for any positive integer q x q := x q · · · x qn . Let P ( Z ) ∈ k [[ x ]][ Z ] be a monic polynomial with coefficients in k [[ x ]]. We denote byNP( P ) the Newton polyhedron of P ( Z ). Let us write P ( Z ) = Z d + X α ∈ Z n ≥ ,j In this note we assume that P ( Z ) = Z d . The associated polyhedron of P , denoted by ∆ P ,is the convex hull of (cid:26) dαd − j | c α,j = 0 (cid:27) + R n ≥ . Note that ∆ P is the projection of NP( P ) from the point (0 , . . . , , d ) on the subspace givenby the first n coordinates. Definition 2.1. Let ω ∈ R n> . For a non zero element b = X α ∈ Z n ≥ b α x α of k [[ x ]] we set ν ω ( b ) := min { α · ω = n X i =1 α i ω i | b α = 0 } ∈ R ≥ and In ω ( b ) := X α | α · ω = ν ω ( b ) b α x α . For such a ω and P ( Z ) ∈ k [[ x ]][ Z ] as before we define ω n +1 ∈ R ≥ by ω n +1 := min { v · ω | v ∈ ∆ P } d ∈ R ≥ . Then we set ω ′ := ( ω, ω n +1 ) and we define ν ω ′ ( P ) := min { α · ω + jω n +1 | c α,j = 0 } = dω n +1 and In ω ′ ( P ) := Z d + X ( α,j ) | ( α,j ) · ω ′ = ν ω ′ ( P ) c α,j x α Z j . This former polynomial is weighted homogeneous for the weights ω ,..., ω n , ω n +1 . Definition 2.2. Let P ( Z ) ∈ k [[ x ]][ Z ] be a monic polynomial of degree d in Z . Thepolynomial P has an orthant associated polyhedron if ∆ P = dγ + R n ≥ for some γ ∈ Q n ≥ .In this case In ω ′ ( P ) does not depend on ω and we denote it by P In , i.e. P In ( x, Z ) := Z d + X ( α,j ) | αd − j = γ c α,j x α Z j . In this case we define P ( Z ) := P In (1 , Z ) = Z d + X ( α,j ) | αd − j = γ c α,j Z j ∈ k [ Z ] . If we write γ = βq , where β ∈ Z n ≥ , q ∈ { , . . . , d } and gcd( β , . . . , β n , q ) = 1, we have that x dβ P ( Z ) = P In ( x q , . . . , x qn , x β Z ) . Here is a picture of the Newton polyhedron of a polynomial having an orthant associatedpolyhedron with n = 2 (thick lines represent the edges of the Newton polyhedron) : x Zx (0 , d )( α, j ) dγ = dαd − j N IRREDUCIBILITY CRITERION FOR POWER SERIES 3 Theorem 2.3. Let us assume that P ( Z ) is irreducible and has an orthant associatedpolyhedron. Then P In ( x, Z ) ∈ k [ x, Z ] is not the product of two coprime polynomials.Proof. Let us assume that P In ( x, Z ) is the product of two coprime polynomials of k [ x, Z ].We denote by P ( x, Z ) and P ( x, Z ) these two polynomials, so we have P In ( x, Z ) = P ( x, Z ) · P ( x, Z ) , and we may assume that they are monic respectively of degree d and d (with d + d = d )since P In ( x, Z ) is monic. Let us write P i ( Z ) := P i (1 , Z ) for i = 1 , 2. Thus we have that P ( Z ) = P ( Z ) · P ( Z ) . Let M = M ( x, Z ) := c x α Z j be a monomial of P ( x, Z ). We have that M ( x q , . . . , x qn , x β Z ) = c x qα + jβ Z j and qα + jβ ≥ ∗ dβ since αd − j ≥ ∗ βq if j < d , where ≥ ∗ denotes the product order on R n ≥ .Thus we have(1) P ( x q , . . . , x qn , x β Z ) = x dβ (cid:0) P ( Z ) + Q ( x, Z ) (cid:1) for some Q ( x, Z ) ∈ ( x ) k [[ x ]][ Z ]. In particular, P ( Z ) + Q ( x, Z ) = P ( Z ) P ( Z ) modulo( x ). Thus by Hensel’s Lemma P ( Z ) + Q ( x, Z ) = e P ( x, Z ) · e P ( x, Z ) , for some monic polynomials e P ( x, Z ) and e P ( x, Z ) ∈ k [[ x ]][ Z ] equal respectively to P ( Z )and P ( Z ) modulo ( x ). So we have that P ( x q , x β Z ) = (cid:16) x d β e P ( x, Z ) (cid:17) · (cid:16) x d β e P ( x, Z ) (cid:17) and h x d i β e P i ( x, Z ) i In = P i ( x q , x β Z ) for i = 1 , . But we have that x d i β e P i ( x, Z ) = R i ( x, x β Z )for some monic polynomials R i ( x, Z ) ∈ k [[ x ]][ Z ] of degree d i . Thus P ( x q , Z ) = R ( x, Z ) · R ( x, Z )and R i In ( x, Z ) = P i ( x q , Z ) ∈ k [ x q , Z ] for i = 1 , 2. Since P In = In ω ′ ( P ) and R i In =In ω ′ ( R i ) for i = 1 , 2, for any ω ∈ R n> we can apply Lemma 2.5 for P = P ( x q , Z ) to seethat R ( x, Z ), R ( x, Z ) ∈ k [[ x q ]][ Z ]. Hence P is not irreducible. (cid:3) Remark 2.4. The key point in the proof of this theorem is the fact that equation (1) issatisfied when P has an orthant associated polyhedron. Lemma 2.5. Let P ∈ k [[ x q ]][ Z ] be a monic polynomial, where q ∈ Z > , and let us assumethat P = R R , where R and R are monic polynomials of k [[ x ]][ Z ] . Let ω ∈ R n> andlet ω ′ be defined as in Definition 2.1. If In ω ′ ( R ) , In ω ′ ( R ) ∈ k [ x q , Z ] and if they arecoprime then R , R ∈ k [[ x q ]][ Z ] .Proof. If char( k ) = p > q = p e m with m ∧ p = 1. If char( k ) = 0 we set m := q and p := 1. Then we define Q := Y R ( ξ x , . . . , ξ n x n , Z ) p e where ( ξ , . . . , ξ n ) runs over the n -uples of m -th roots of unity in an algebraic closureof k . Then Q ∈ k [[ x q ]][ Z ] and In ω ′ ( Q ) = In ω ′ ( R ) m n p e . Thus In ω ′ ( R ) and In ω ′ ( Q )are coprime. Hence the greatest common divisor of P and Q in k (( x ))[ Z ] is R . Butthe greatest common divisor does not depend of the base field so R is also the greatest GUILLAUME ROND AND BERND SCHOBER common divisor of P and Q in k (( x q ))[ Z ] hence R ∈ k [[ x q ]][ Z ]. By symmetry we alsoget R ∈ k [[ x q ]][ Z ]. (cid:3) Corollary 2.6. Let us assume that P ( Z ) = Z d + a Z d − + . . . + a d ∈ k [[ x ]][ Z ] is irreducible.Then we have the following properties:i) If P ( Z ) has an orthant associated polyhedron the convex hull of Supp(In ω ′ ( P )) is a segment joining (0 , d ) to ( dγ, , and dγ is the initial exponent of a d for thevaluation ν ω .ii) If P ( Z ) has an orthant associated polyhedron let u ∈ Z n +1 be the primitive vectorsuch that mu = ( − dγ, d ) for some m ∈ N , and set y := ( x, Z ) . Then we can write P In ( x, Z ) = x dγ Q ( y u ) where Q ( T ) ∈ k [ T ] is not the product of two coprime polynomials. In particular, Q ( T ) has only one root in an algebraic closure of k .iii) If the Newton polyhedron of P ( Z ) has no compact face of dimension > then P ( Z ) has an orthant associated polyhedron and its Newton polyhedron has onlyone compact face of dimension one which is the segment of i).Proof. If P In ( x, 0) = 0 then Z divides P In ( x, Z ). But by Theorem 2.3 P In ( x, Z ) is not theproduct of two coprime polynomials thus P In ( x, Z ) = Z d . This contradicts the fact that P In ( x, Z ) has a non zero monomial of the form x α Z j for j < d . Hence P In ( x, = 0 and i ) is proven.We can write P In ( x, Z ) = Z d + d − X j =0 c ( d − j ) γ,j x ( d − j ) γ Z j . So we have that P In ( x, Z ) = x dγ Z d x dγ + d − X j =0 c ( d − j ) γ,j Z j x jγ ! . By i ) we have that dγ ∈ Z n ≥ . This implies that jγ ∈ Z n ≥ as soon as c ( d − j ) γ,j = 0. Forany such j , let i ≥ iu = ( − jγ, j ) . Then i ∈ Z ≥ since u is primitive.Thus P In ( x, Z ) = x dγ ( y mu + P i Then the polynomials P ( x, Z ) := x d γ Q ( y u ) and P ( x, Z ) := x d γ Q ( y u ) are coprimewhich contradicts Theorem 2.3. Thus ii ) is proven.Let us assume that the Newton polyhedron of P ( Z ) does not have an orthant associatedpolyhedron. This means that ∆ P has at least two distinct vertices denoted by γ and γ such that the segment [ γ , γ ] is included in the boundary of ∆ P . Thus the Newtonpolyhedron of P has at least three different vertices a := (0 , d ), b := ( d − jd γ , j ) and c := ( d − kk γ , k ). Since a , b , c are vertices of NP( P ) the triangle delimitated by these threepoints is a face of NP( P ) so the Newton polyhedron of P has at least one face of dimensiontwo. (cid:3) An example concerning compact faces of dimension > n = 2 and let us replace the variables ( x , x ) by ( x, y ) for simplicity. We set P ( Z ) := Z − ( x − y ) + y = ( Z − x + y )( Z + x − y ) + y seen as a polynomial of k [[ x, y ]][ Z ] where k is an algebraically closed field of characteristicdifferent from 2.We will show that P does not have an orthant associated polyhedron, since ∆ P has twodifferent vertices. On the other hand, we will prove that P ( Z ) is irreducible while forevery ω ∈ R > the polynomial In ω ′ ( P ) is always the product of two coprime monic poly-nomials. This shows that Theorem 2.3 cannot be extended to polynomials without anorthant associated polyhedron.The Newton polyhedron of P ( Z ) is the convex hull of { (6 , , , (0 , , , (0 , , } + R ≥ . The associated polyhedron ∆ P of P ( Z ) is the convex hull of { (6 , , (0 , } + R ≥ and has two vertices v = (6 , 0) and u = (0 , ω ∈ R > , if 6 ω < ω thenIn ω ′ ( P ) = Z − x = ( Z − x )( Z + x ) . If 6 ω > ω then we have thatIn ω ′ ( P ) = Z − y = ( Z − y )( Z + y ) . If 6 ω = 10 ω we have thatIn ω ′ ( P ) = Z − ( x − y ) = ( Z − x + y )( Z + x − y ) . Thus in all cases In ω ′ ( P ) is the product of two coprime polynomials (since char( k ) = 2).On the other hand, P ( Z ) is irreducible since ( x − y ) − y is not a square in k [[ x, y ]]. References [ACLM1] E. Artal Bartolo, P. Cassou-Nogu`es, I. Luengo, A. Melle Hern´andez, On ν -quasi-ordinarypower series: factorization, Newton trees and resultants, Topology of algebraic varieties and sin-gularities (Jaca, 2009), Contemp. Math., vol. 538, Amer. Math. Soc., Providence, RI, 2011, pp.321-343.[ACLM2] E. Artal Bartolo, P. Cassou-Nogu`es, I. Luengo, A. Melle Hern´andez, Quasi-ordinary singular-ities and Newton trees, Mosc. Math. J. , , (2013), no. 3, 365-398.[GBGP] E. R. Garc´ıa Barroso, P. D. Gonz´alez-P´erez, Decomposition in bunches of the critical locus ofa quasi-ordinary map, Compos. Math. , , (2005), no. 2, 461-486.[MS] H. Mourtada, B. Schober, A polyhedral characterization of quasi-ordinary singularities,arXiv:1512.07507. GUILLAUME ROND AND BERND SCHOBER Guillaume Rond, Aix-Marseille Universit´e, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Mar-seille, France E-mail address : [email protected] Bernd Schober, Institut f¨ur Algebraische Geometrie, Leibniz Universit¨at Hannover, Welfengarten1, 30167 Hannover, Germany E-mail address ::