Computing the equisingularity type of a pseudo-irreducible polynomial

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Computing the equisingularity type of a pseudo-irreducible polynomial

Adrien Poteaux, Martin Weimann

To cite this version:

Adrien Poteaux, Martin Weimann. Computing the equisingularity type of a pseudo-irreducible poly-

nomial. Applicable Algebra in Engineering, Communication and Computing, Springer Verlag, 2020,

31, pp.435-460. �10.1007/s00200-020-00451-x�. �hal-02354930v2�

(will be inserted by the editor)

Computing the equisingularity type of a pseudo-irreducible polynomial

Adrien Poteaux · Martin Weimann

Received: date / Accepted: date

Abstract Germs of plane curve singularities can be classified accordingly to their equisingularity type. For singularities overC, this important data coincides with the topological class. In this paper, we characterise a family of singularities, containing irreducible ones, whose equisingularity type can be computed in an expected quasi- linear time with respect to the discriminant valuation of a Weierstrass equation.

Keywords Polynomials·Plane curves·Equisingularity·Pseudo-irreduibility· Approximate roots

Mathematics Subject Classification (2010) 14Q20·12Y05·13P05·68W30

1 Introduction

Equisingularity is the main notion of equivalence for germs of plane curves. It was developed in the 60’s by Zariski over algebraically closed fields of characteristic zero in [29–31] and generalised in arbitrary characteristic by Campillo [2]. This concept is of particular importance as for complex curves, it agrees with the topological equiv- alence class [29]. As illustrated by an extensive literature (see e.g. the book [9] and the references therein), equisingularity plays nowadays an important role in various active fields of singularity theory (resolution, equinormalisable deformation, mod- uli problems, analytic classification, etc). It is thus an important issue of computer

A. Poteaux

CRIStAL, CNRS UMR 9189, Universit´e de Lille, Bˆatiment Esprit, 59655 Villeneuve d’Ascq, France E-mail: adrien.poteaux@univ-lille.fr

M. Weimann

Permanent address :LMNO, CNRS UMR 6139, Universit´e de Caen-Normandie, BP 5186, 14032 Caen Cedex, France

Present address :GAATI, EA 3893, Universit´e de Polyn´esie Franc¸aise, BP 6570, 98702 Faa’a, Polyn´esie Franc¸aise

E-mail: martin.weimann@upf.pf

algebra to design efficient algorithms for computing the equisingularity type of a sin- gularity. This paper is dedicated to characterise a family of reduced germs of plane curves, containing irreducible ones, for which this task can be achieved in quasi-linear time with respect to the discriminant valuation of a Weierstrass equation.

Main result. We say that two germs of reduced plane curves are equisingular if there is a one-to-one correspondence between their branches which preserves the charac- teristic exponents and the pairwise intersection multiplicities (see e.g. [2,3,27] for other equivalent definitions). This equivalence relation leads to the notion of equi- singularity type of a singularity. In this paper, we consider a square-free Weierstrass polynomialF∈K[[x]][y]of degreed, withKa perfect field of characteristic zero or greater thand¹. Under such an assumption, the Puiseux series ofF are well defined and allow to determine the equisingularity type of the germ(F,0)(the case of small characteristic requires Hamburger-Noether expansions [2]). In particular, it follows from [22] that we can compute the equisingularity type in an expectedO˜(dδ)²op- erations over K, where δ stands for the valuation of the discriminant ofF. IfF is irreducible, it is shown in [23] that we can reach the lower complexityO˜(δ)thanks to the theory of approximate roots. A natural question arises : can we reach this quasi- optimal complexityO˜(δ)for a larger class of reducible polynomials ?

We say thatFispseudo-irreducible³orbalanced⁴if all its absolutely irreducible factors have the same set of characteristic exponents and the same set of pairwise intersection multiplicities, see Section2. Irreducibility over any algebraic field ex- tension ofKimplies pseudo-irreducibility by a Galois argument, but the converse does not hold. As a basic example, the Weierstrass polynomialF= (y−x)(y−x²)is pseudo-irreducible, but is obviously reducible (see Example1for more). We prove:

Theorem 1. There exists an algorithm which tests if F is pseudo-irreducible with an expectedO˜(δ)⁵operations overK. If F is pseudo-irreducible, the algorithm com- putes alsoδand the number of absolutely irreducible factors of F together with their sets of characteristic exponents and sets of pairwise intersection multiplicities. In particular, it computes the equisingularity type of the germ(F,0).

Remark1. The algorithm contains a Las Vegas subroutine for computing primitive elements in residue rings; however it should become deterministic thanks to [13].

Our algorithm does not perform univariate irreducible factorisation, but uses only square-free factorisation instead. However, for a given field extensionLofK, we can also compute the degrees, residual degrees and ramification indices of the irreducible factors ofF inL[[x]][y]by performing an extra univariate factorisation of degree at mostdoverL. Our long term objective being fast factorisation inK[[x]][y], we extend in Section5.3the definition of pseudo-irreducibility to non Weierstrass polynomials,

1 Our results still hold under the weaker assumption that the characteristic ofKdoes not divided.

2 As usual, the notationO˜()hides logarithmic factors

3 The terminologypseudo-irreducibleis also used in [14] to design polynomials which cannot be fac- tored into comaximal polynomials. These two notions are not related to each other.

4 In the sequel, we rather use the terminologybalancedand give an alternative definition of pseudo- irreducibility based on a Newton-Puiseux type algorithm. Both notions agree from Theorem2.

5 Note thatFbeing Weierstrass, we haved≤δandδlog(d)∈O˜(δ).

taking into account all germs of curves defined byFalong the linex=0. In this more general setting, we get a complexityO˜(δ+d)⁶.

Main tools. We generalise the irreducibility test obtained in [23], which is itself a generalisation of Abhyankhar’s absolute irreducibility criterion [1], based on the the- ory of approximate roots. The main idea is to compute recursively some suitable ap- proximate rootsψ0, . . . ,ψ_gofFof strictly increasing degrees such thatFis pseudo- irreducible if and only if we reachψ_g=F. At stepk, we compute the(ψ₀, . . . ,ψ_k)- adic expansion ofF from which we can construct a generalised Newton polygon. If the corresponding boundary polynomial ofF is not pseudo-degenerate (Definition 4), then F is not pseudo-irreducible. Otherwise, we deduce the degree of the next approximate rootψk+1that has to be computed.

Our algorithm is optimised in the sense that it computes only the minimal amount of information required for testing pseudo-irreducibility and computing the equisin- gular type (roughly speaking, it computes only the initial terms of the Puiseux series y_i−y_jwherey_i,y_jrun over the roots ofF, see Remark6).

Differences and additional interest with the irreducibility test of [23]. The key differ- ence when compared to the irreducibility test developed in [23] is that we may allow the successive generalised Newton polygons to have several edges, although no split- tings and no Hensel liftings are required. Except this slight modification, most of the algorithmic considerations have already been studied in [23] and the main inter- est of this paper is more of a theoretical nature, focused on two main points: proving that pseudo-degeneracy is the right condition for characterising pseudo-irreducibility, and giving formulas for the intersection multiplicities and characteristic exponents in terms of the underlying edge data sequence. This is our main Theorem2.

Additionnaly to the statements of our main theorems, their proofs enlighten the important following fact : a polynomial F is pseudo-irreducible if and only if its equisingularity type can be read on a suitable sequence of approximate roots of F. As a consequence, the underlying algorithm relies only on a few “black box” functions.

In particular, the dynamic evaluation inherent to arithmetic operations in the towers of residue rings is in such a case reduced to univariate computations (the remaining tasks being simply Newton iterations and multiplications inK[[x]][y]), which makes this algorithm easy to implement. If F is not pseudo-irreducible, then we need at some point to compute a factorisation F=GH (up to some suitable precision) in order to apply the approximate roots strategy on each discovered factor G,H. In [22], it is shown that these factorisation steps costO˜(dδ), assuming a divide and conquer strategy. The underlying algorithm is much more tricky to implement as it uses dynamic evaluation for bivariate computations (for instance, the extension of the half-gcd [16, Algorithm 1] in the context of dynamic evaluation, a key point of the divide and conquer strategy, is not available in any classical library for the moment). These considerations show that, up to our knowledge, pseudo-irreducible polynomials consitute the largest class of polynomials whose equisingularity type can be efficiently computed with quasi-linear complexityO˜(δ)in the current state

6 WhenFis Weierstrass, we haved≤δ. Otherwise, we might haved∈/O˜(δ).

of the art. As such, they deserve to be considered as elementary bricks for a fast and easy-to-implement algorithm which computes the equisingularity type of a general germ of curve. This larger program is an ongoing work.

Related results. Computing the equisingularity type of a plane curve singularity is a classical topic for which both symbolic and numerical methods exist. A classical approach is derived from the Newton-Puiseux algorithm, as a combination of blow- ups (monomial transforms and shifts) and Hensel liftings. This approach allows to compute the roots ofF - represented as fractional Puiseux series - up to an arbitrary precision, from which the equisingularity type of the germ(F,0)can be deduced (see e.g. Theorem2for precise formulas). The Newton-Puiseux algorithm has been stud- ied by many authors (see e.g. [4,5,19–23,26,28] and the references therein). Up to our knowledge, the best current arithmetic complexity was obtained in [22], comput- ing the singular parts of all Puiseux series abovex=0 - hence the equisingularity type of all germs of curves defined byFalong this line - in an expectedO˜(dδ)operations overK. Here, we get rid of thed factor for pseudo-irreducible polynomials, gener- alising the irreducible case considered in [23]. For complex curves, the equisingular- ity type agrees with the topological class and there exists other numerical-symbolic methods of a more topological nature (see e.g. [10–12,17,25] and the references therein). The main results of this paper and the irreducibility test in [23] are grouped together in a long preprint [24]. The length of this preprint finally led us to publish it into two separate parts.

Organisation. We define balanced polynomials in Section 2. Section 3 introduces the notion of pseudo-degeneracy. This leads to an alternative definition of a pseudo- irreducible polynomial, based on a Newton-Puiseux type algorithm. In Section 4, we prove that being balanced is equivalent to being pseudo-irreducible and we give explicit formulas for characteristic exponents and intersection multiplicities in terms of edge data (Theorem2). In the last Section5, we design a pseudo-irreducibility test based on approximate roots with quasi-linear complexity, thus proving Theorem1.

We finally illustrate our method on various examples.

2 Balanced polynomials

Let us fixF ∈K[[x]][y]a Weierstrass polynomial defined over a perfect field Kof characteristic zero or greater thand=deg(F). We denote by(F,0)the germ of the plane curve defined byF at the origin of the affine planeK². We say thatF is abso- lutely irreducible if it is irreducible inK[[x]][y]. The germs of curves defined by the absolutely irreducible factors ofFare called the branches of the germ(F,0).

2.1 Characteristic exponents

We assume here thatF is absolutely irreducible. As the characteristic ofKdoes not divided, there exists a unique seriesS(T) =∑c_iTⁱ∈K[[T]]such thatF(T^d,S(T)) =

0. The pair(T^d,S(T))is the classical Puiseux parametrisation of the branch(F,0).

Thecharacteristic exponentsofFare defined as

β₀=d, β_k=min(i s.t.c_i6=0,gcd(β₀, . . . ,βk−1)6 |i), k=1, . . . ,g, wheregis the least integer for which gcd(β₀, . . . ,βg) =1 (characteristic exponents are sometimes referred to the rational numbersβi/d in the literature). These are the exponentsifor which a non trivial factor of the ramification index is discovered. It is well known that the data

C(F) = (β₀;β1, . . . ,βg)

determines the equisingularity type of the germ(F,0), see e.g. [29]. Conversely, the Weierstrass equations of two equisingular germs of curves which are not tangent to the x-axis have same characteristic exponents [3, Corollary 5.5.4]. If tangency occurs, we rather need to consider the characteristic exponents of the local equation obtained after a generic change of local coordinates, which form a complete set of equisingular (hence topological if K=C) invariants. The set C(F)and the set of generic characteristic exponents determine each others assuming that we are given the contact orderβ0 withx-axis ([18, Proposition 4.3] or [3, Corollary 5.6.2]). It is well known that a data equivalent to C(F)is given by the semi-group ofF, and that this semi-group admits the intersection multiplicities ofFwith its characteristic approximate rootsψ−1,ψ0, . . . ,ψgas a minimal system of generators (see Section5.1 and [3, Corollaries 5.8.5 and 5.9.11]).

2.2 Intersection sets

If we want to determine the equisingularity type of a reducible germ(F,0), we need to consider also the pairwise intersection multiplicities between the absolutely irre- ducible factors ofF. The intersection multiplicity between two coprime Weierstrass polynomialsG,H∈K[[x]][y]is defined as

(G,H)₀:=vx(Res_y(G,H)) =dim

K

K[[x]][y]

(G,H) , (1)

where Res_ystands for the resultant with respect toyandv_xis the usualx-valuation.

The right hand equality follows from classical properties of the resultant. Suppose thatF has (distinct) absolutely irreducible factorsF₁, . . . ,F_f. We introducethe inter- section setsofF, defined fori=1, . . . ,f as

Γi(F):= ((F_i,Fj)₀,1≤j≤f,j6=i).

By convention, we take into account repetitions,Γi(F)being considered as an un- ordered list with cardinality f−1. IfF is Weierstrass, the equisingular type (hence the topological class ifK=C) of the germ(F,0)is uniquely determined by the char- acteristic exponents and the intersections sets of the branches ofF[32]. Note that the set C(F_i)only depends onF_iwhileΓi(F)depends onF.

2.3 Balanced polynomials

Definition 1. We say that a square-free Weierstrass⁷polynomialF∈K[[x]][y]isbal- ancedif C(F_i) =C(F_j)andΓ_i(F) =Γ_j(F)for alli,j. In such a case, we denote simply these sets by C(F)andΓ(F).

Thus, if F is balanced, its branches are equisingular and have the same set of pairwise intersection multiplicities. The converse holds if no branch is tangent to the x-axis or all branches are tangent to thex-axis.

Example 1. Let us illustrate this definition with some basic examples. Note that the second and third examples show in particular that no condition implies the other in Definition1.

1. IfF∈K[[x]][y]is irreducible, a Galois argument shows that it is balanced (fol- lows from Theorem2below). The converse doesn’t hold:F= (y−x)(y+x²)is reducible, but it is balanced. This example also shows that being balanced does not imply the Newton polygon to be straight.

2. F= (y²−x³)(y²+x³)(y²+x³+x⁴)is not balanced. It has 3 absolutely irreducible factors with same sets of characteristic exponents C(F_i) = (2; 3)for all i, but Γ1(F) = (6,6)whileΓ2(F) =Γ3(F) = (6,8).

3. F= (y−x−x²)(y−x+x²)(y²−x³)is not balanced. It has 3 absolutely irreducible factors with same sets of pairwise intersection multiplicitiesΓi(F) = (2,2), but C(F₁) =C(F₂) = (1)while C(F₃) = (2; 3).

4. F= (y²−x²)²−2x⁴y²−2x⁶+x⁸has four absolutely irreducible factors, namely F₁=y+x+x²,F₂=y+x−x²,F₃=y−x+x²andF₄=y−x−x². We have C(F_i) = (1)andΓi(F) = (1,1,2)for allisoFis balanced. Note that this example shows that being balanced does not imply that all factors intersect each others with the same multiplicity.

5. F= (y²−x³)(y³−x²)is not balanced. However, it defines two equisingular germs of plane curves (but one is tangent to thex-axis while the other is not).

Noether-Merle’s Formula. IfF,G∈K[[x]][y]are two irreducible Weierstrass polyno- mials of respective degreesd_Fandd_G, their intersection multiplicity(F,G)₀is closely related to the characteristic exponents(β₀, . . . ,βg)ofF. Let us denote by

Cont(F,G):=dFmax vx(y−y⁰)|F(y) =0,G(y⁰) =0

(2) the contact orderof the branchesF andGand letκ =max{k|Cont(F,G)≥β_k}.

Then Noether-Merle’s formula [15, Proposition 2.4] states (F,G)₀=d_G

d_F

∑

k≤κ

(E_k−1−E_k)β_k+EκCont(F,G)

!

, (3)

whereE_k:=gcd(β₀, . . . ,β_k). A proof can be found in [18, Proposition 6.5] (and refer- ences therein), where a formula is given in terms of the semi-group generators, which turns out to be equivalent to (3) thanks to [18, Proposition 4.2]. Note that the original proof in [15] assumes that the germsFandGare transverse to thex-axis.

7 We can extend this definition to non-Weierstrass polynomials, see Section5.3.

3 Pseudo-irreducible polynomials

3.1 Pseudo-degenerate polynomials.

We first recall classical definitions that play a central role for our purpose, namely the Newton polygon and the residual polynomial. We will have to work over vari- ous residue rings isomorphic to some direct product of field extensions of the base field K. Let A=L0⊕ · · · ⊕Lr be such a ring. If S =∑c_ixⁱ ∈A[[x]], we define v_x(S) =min(i,c_i6=0)with conventionv_x(0) = +∞. Note that in contrast to usual val- uations, we havev_x(S₁S₂)≥v_x(S₁) +v_x(S₂)and strict inequality might occur since Ais allowed to contain zero divisors. We letA^×stands for the group of units ofA.

In the following definitions, we assume thatF ∈A[[x]][y]is a Weierstrass poly- nomial and we letF=∑^di=0a_i(x)yⁱ=∑i,ja_{i j}x^jyⁱ. The support Supp(F)⊂N²is the set of exponents(i,j)for whicha_{i j}6=0.

Definition 2. TheNewton polygonofFis the lower convex hullN(F)of its support.

The unique edge with vertice(d,0)is called the lower edge, denoted byN₀(F).

The lower edge has equationmi+q j=lfor some uniquely determined coprime positive integersq,mandl∈N. We say for short thatN₀(F)has slope(q,m), with convention(q,m) = (1,0)if the Newton polygon ofFis reduced to a point.

Definition 3. We call ¯F:=∑(i,j)∈N₀(F)a_{i j}x^jyⁱthelower boundary polynomialofF. We say that a polynomialP∈A[Z]issquare-freeif its images under the natural morphismsA→Liare square-free (in the usual sense over a field).

Definition 4. We say thatF∈A[[x]][y]ispseudo-degenerateif there existsN∈N andP∈A[Z]monic and square-free such that

F¯=

P y^q

x^m

x^mdeg(P) N

, (4)

with moreoverP(0)∈A^×ifq>1. We callPtheresidual polynomialofF. The tuple (q,m,P,N)is theedge dataofF.

Remark2. In practice, we check pseudo-degeneracy as follows. Ifqdoes not divide d, thenF is not pseudo-degenerate. Ifq dividesd, thenq|i for all(i,j)∈ N₀(F) as(d,0)∈ N₀(F)by assumption. Hence we may considerQ=∑(i,j)∈N₀(F)ai jZ^i/q∈ A[Z]andFis pseudo-degenerate if and only ifQ=P^N for some square-free polyno- mialPsuch thatP(0)∈A^×ifq>1.

Remark 3. If q>1, the extra condition P(0)∈A^×implies that N(F) is straight (i.e. it has a unique edge). Ifq=1, we allowP(0)to be a zero-divisor (in contrast to Definition 4 of quasi-degeneracy in [23]), in which caseN(F)may have several edges. Note that ifF is pseudo-degenerate, ¯F is the power of a square-free quasi- homogeneous polynomial, but the converse doesn’t hold (case4below).

Example 2. a

1. LetF = (y²−x²)²(y−x²)(y−x³). Then N(F)has three edges, the lower one of slope(q,m) = (1,1). We get ¯F= (y³−x²y)²andQ= (Z³−Z)². Hence,Fis pseudo-degenerate, withP=Z³−ZandN=2.

2. LetF = (y²−x²)²(y−x²). ThenN(F)has two edges, the lower one of slope (q,m) = (1,1). We get ¯F=y(y²−x²)²andQ=Z(Z²−1)²is not a power of a square-free polynomial. Hence,Fis not pseudo-degenerate.

3. LetF = (y²−x³)²(y−x⁴). ThenN(F)has two edges, the lower one of slope (q,m) = (2,3). Asqdoes not divided=5,Fis not pseudo-degenerate.

4. LetF= (y²−x³)²(y−x⁴)². ThenN(F)is straight of slope(q,m) = (2,3). Here qdividesd=6. We get ¯F=y²(y²−x³)²which is a power of a square-free poly- nomial. However,Q=Z(Z−1)²is not. Hence,Fis not pseudo-degenerate.

5. LetF= (y²−x³)²(y²−x⁴)². ThenN(F)has two edges, the lower one of slope (q,m) = (2,3). Hereqdividesd=8. We get ¯F= (y⁴−y²x³)²andQ= (Z²− Z)²is the power of the square-free polynomialP=Z²−Z. However,q>1 and P(0) =0 soFis not pseudo-degenerate.

Note that we could also treat cases 3, 4 and 5 simply by using Remark3:q>1 andN(F)not straight imply thatFis not pseudo-degenerate.

The next lemma allows to associate to a pseudo-degenerate polynomialF a new Weierstrass polynomial of smaller degree, generalising the usual case (e.g. [5, Section 4] or [21, Proposition 3]) to the case of product of fields.

Lemma 1. Suppose that F is pseudo-degenerate with edge data(q,m,P,N)and de- note(s,t)the unique positive integers such that s q−t m=1,0≤t<q. Let z be the residue class of Z in the ringAP:=A[Z]/(P(Z))and`:=deg(P). Then

F(z^tx^q,x^m(y+z^s)) =x^qm`NU G, (5) where U,G∈AP[[x]][y], U(0,0)∈A^×_P and G is a Weierstrass polynomial of degree N dividing d. Moreover, if F6=y^dand F has no terms of degree d−1, then N<d.

Proof. As F is pseudo-degenerate, its lower edgeN₀(F) has equation mi+q j= qm`Nand we may writeF=F¯+Q, with ¯Fas in (4) and where Supp(Q)lies strictly aboveN<sub>0</sub>(F). In particular,vx(Q(z<sup>t</sup>x<sup>q</sup>,x<sup>m</sup>(y+z<sup>s</sup>)))>qm`N. Combined with (4), we get that ˜F(x,y):=x^−qm`NF(z^tx^q,x^m(y+z^s))satisfies ˜F ∈AP[[x]][y] and ˜F(0,y) = R(y)^N whereR(y) =P((y+z^s)^q/z^tm). In particular, we haveR(0) =P(z) =0 and R⁰(0) =qz^1−sP⁰(0). AsPis square-free and the characteristic ofAdoes not divide deg(P)by assumption, we haveP⁰(0)∈A^×. Asqz^1−s∈A^×_P (ifq>1, the assumption P(0)∈A^×impliesPandZcoprime, that isz∈A^×_P; ifq=1, thens=1), it follows thatR⁰(0)∈A^×_P. We deduce that ˜F(0,y) =y^NS(y)wherey^NandS(y)are coprime in AP[y]. We conclude thanks to the Weierstrass preparation theorem thatFfactorises as in (5). Note thatN|dby (4). IfN=d, then (4) forces ¯F= (y+αx^m)^dfor someα∈A. Ifα=0, thenN(F)is reduced to a point and we must haveF=y^d. Ifα 6=0, the coefficient ofy^d−1inFisdαx^{m d}+h.o.t, hence is non zero since the characteristic of Adoes not divided.

Remark4. AsP∈A[Z]is square-free, the ringAP=A[Z]/(P(Z))is still isomorphic to a direct product of perfect fields thanks to the Chinese Reminder Theorem. Note also thatz^t is invertible: ifzis a zero divisor, we must haveq=1 so thatt=0 and z^t=1.

3.2 Pseudo-irreducible polynomials

The definition of a pseudo-irreducible polynomial is based on a variation of the clas- sical Newton-Puiseux algorithm. Thanks to Lemma1, we associate toF a sequence of Weierstrass polynomialsH0, . . . ,H_gof strictly decreasing degreesN0, . . . ,N_gsuch thatH_kis pseudo-degenerate ifk<gand such that eitherHgis not pseudo-degenerate orNg=1. We proceed recursively as follows:

Rank k=0. LetN₀=d,K0=K,c₀(x):=−Coef(F,y^N0−1)/N₀and

H₀(x,y):=F(x,y+c₀(x))∈K0[[x]][y]. (6) ThenH₀is a new Weierstrass polynomial of degreeN₀with no terms of degreeN₀−1.

IfN₀=1 orH₀is not pseudo-degenerate, we letg=0.

Rank k>0. Suppose givenKk−1a direct product of fields extension ofKandH_k−1∈ Kk−1[[x]][y]a pseudo-degenerate Weierstrass polynomial of degreeN_k−1>1, with no terms of degreeNk−1−1. Denote by(q_k,m_k,P_k,N_k)its edge data and`_k:=deg(P_k).

AsP_k is square-free, the ringKk:=Kk−1[Z_k]/(P_k(Z_k))is again (isomorphic to) a direct product of fields. We let z_k ∈Kk be the residue class ofZ_k and (s_k,t_k) the unique positive integers such thats_kq_k−t_km_k=1, 0≤t_k<q_k. AsHk−1is pseudo- degenerate, we deduce from Lemma1that

Hk−1(z^t_k^kx^qk,x^mk(y+z^s_k^k)) =x^qkmk`kNkU_kG_k, (7) whereU_k(0,0)∈K^×_k andG_k∈Kk[[x]][y]is a Weierstrass polynomial of degreeN_k. Lettingc_k:=−Coef(G_k,y^Nk−1)/N_k, we define

H_k(x,y) =G_k(x,y+c_k(x))∈Kk[[x]][y]. (8) It is a degreeN_kWeierstrass polynomial with no terms of degreeN_k−1.

The N_k-sequence stops. We have the relationsN_k=q_k`_kNk−1. AsHk−1 is pseudo- degenerate with no terms of degreeNk−1−1, we haveN_k<Nk−1by Lemma1. Hence the sequence of integersN₀, . . . ,N_kis strictly decreasing and there exists a smallest indexgsuch that eitherN_g=1 (andH_g=y), eitherN_g>1 andH_gis not pseudo- degenerate. We collect the edge data of the polynomialsH₀, . . . ,Hg−1in a list

Data(F):= ((q₁,m₁,P₁,N₁), . . . ,(q_g,m_g,P_g,N_g)).

Note thatm_k>0 for all 1≤k≤g. We include theN_k’s in the list for convenience (they could be deduced from the remaining data via the relationsN_k=Nk−1/q_k`_k).

Definition 5. We say thatFispseudo-irreducibleifN_g=1.

4 Pseudo-irreducible is equivalent to balanced.

We prove here our main result, Theorem 2: a square-free Weierstrass polynomial F ∈K[[x]][y]is pseudo-irreducible if and only if it is balanced, in which case we compute characteristic exponents and intersection sets of the irreducible factors.

4.1 Notation and main results.

We use the notations of Section3; in particular(q₁,m₁,P₁,N₁), . . . ,(q_g,m_g,P_g,N_g) denote the edge data ofF. We definee_k:=q₁· · ·q_k (current index of ramification), e:=e_g, ˆe_k:=e/e_kand in an analogous way f_k:=`<sub>1</sub>· · ·`_k(current residual degree),

f :=fgand ˆf_k:=f/f_k. For allk=1, . . . ,g, we define

B_k=m1eˆ1+· · ·+m_keˆ_k and M_k=m1eˆ0eˆ1+· · ·+m_keˆk−1eˆ_k (9) and we letB₀=e. These are positive integers related by the formula

M_k=

k i=1

∑

(eˆi−1−eˆ_i)B_i+eˆ_kB_k. (10) Note that 0<B₁≤ · · · ≤B_g⁸andB₀≤B_g. We haveB₀≤B₁if and only ifq₁≤m₁, if and only ifF=0 is not tangent to thex-axis at the origin. We check easily that

ˆ

e_k=gcd(B₀, . . . ,B_k). In particular, gcd(B₀, . . . ,B_g) =1.

Theorem 2. A Weierstrass polynomial F∈K[[x]][y]is balanced if and only if it is pseudo-irreducible. In such a case, F has f irreducible factors inK[[x]][y], all with degree e, and

1. C(F) = (B₀;B_k|q_k>1)- so C(F) = (1)if q_k=1for all k.

2. Γ(F) = (M_k|`_k>1), where M_kappears fˆk−1−fˆ_ktimes.

Taking into account repetitions, the intersection set has cardinality∑^g_k=1(fˆk−1− fˆ_k) =f−1, as required. It is empty if and only ifF is absolutely irreducible.

Corollary 1. Let F∈K[[x]][y]be a balanced Weierstrass polynomial. Then, the dis- criminant of F has valuation

δ=f

∑

`_k>1

(fˆk−1−fˆ_k)M_k+

∑

qk>1

(eˆk−1−eˆ_k)B_k

!

and the discriminants of the absolutely irreducible factors of F all have the same valuation∑q_k>1(eˆk−1−eˆ_k)B_k.

Proof. (of Corollary1)WhenFis balanced, it has f irreducible factorsF1, . . . ,F_f of same degreee, with discriminant valuations sayδ₁, . . . ,δ_f. The multiplicative prop- erty of the discriminant gives the well-known formula

δ=

∑

1≤i≤f

δ_i+

∑

1≤i6=j≤f

(F_i,F_j)₀. (11) Lety₁, . . . ,y_ebe the roots ofF_i. Thanks to [27, Proposition 4.1.3 (ii)] combined with point 1 of Theorem2, we deduce that for each fixeda=1, . . . ,e, the list(v_x(y_a− y_b),b6=a)consists of the valuesB_k/erepeated ˆek−1−eˆ_ktimes fork=1, . . . ,g. Since δ_i=∑1≤a6=b≤ev_x(y_a−y_b), we deduce thatδ₁=· · ·=δ_f =∑_qk>1(eˆk−1−eˆ_k)B_k. The formula forδfollows directly from (11) combined with point 2 of Theorem2.

8 We may allowm₁=B₁=0 when considering non Weierstrass polynomials, see Section5.3.

The remaining part of this section is dedicated to the proof of Theorem2. It is quite technical, but has the advantage to be self-contained. We first establish the re- lations between the (pseudo)-rational Puiseux expansions and the classical Puiseux series ofF(Section4.2). This allows us to deduce the characteristic exponents and the intersection sets of a pseudo-irreducible polynomial (thanks to Noether-Merle’s for- mula), proving in particular that pseudo-irreducible implies balanced (Section4.3).

We prove the more delicate reverse implication in Section4.4.

4.2 Pseudo-rational Puiseux expansion.

Keeping the notations of Section3, letπ0(x,y) = (x,y+c₀(x))andπk=πk−1◦σk

where

σk(x,y):= (z^t_k^kx^qk,x^mk(y+z^s_k^k+c_k(x))) (12) fork≥1. It follows from equalities (6), (7) and (8) that

π_k^∗F=x^vx(πk∗F)V_kH_k∈Kk[[x,y]] (13) for someV_ksuch thatV_k(0,0)∈K^×_k. We deduce from (12) that

π_k(x,y) = (µ_kx^ek,α_kx^rky+S_k(x)), (14) whereµ_k,α_k∈K^×_k,r_k∈NandS_k∈Kk[[x]]satisfiesv_x(S_k)≤r_k. Following [22], we call the parametrisation

(µ_kT^ek,S_k(T)):=π_k(T,0)

a pseudo-rational Puiseux expansion (pseudo-RPE for short). Its ring of definition equals the current residue ringKk, which is a reduced zero-dimensionalK-algebra of degree f_koverK. WhenF is irreducible, theKk’s are fields and the parametrisation πk(T,0)allows to compute the Puiseux series ofFtruncated up to precision_e^rk

k, which increases withk[22, Section 3.2]. We show here that the same conclusion holds when F is pseudo-irreducible, taking care of possible zero-divisors inKk. To this aim, we prove by induction an explicit formula forπ_k(T,0). We need further notations.

Exponents data. For all 0≤i≤k≤g, we defineQk,i=qi+1· · ·q_kwith convention Qk,k=1 and let

B_k,i=m₁Q_k,1+· · ·+m_iQ_k,i

with conventionB_k,0=0. Note thatQ_i,0=e_i,Q_g,i=eˆ_iandB_g,i=B_ifor alli≤g. We have the relationsQ_k+1,i=q_k+1Q_k,iandB_k+1,i=q_k+1B_k,ifor alli≤kandB_k+1,k+1= q_k+1B_k,k+m_k+1.

Coefficients data. For all 0≤i≤k≤g, we define µ_k,i :=z^t_i+1^i+1Qi,i· · ·z^t_k^kQk−1,i with conventionµ_k,k=1 and let

α_k,i:=µ_k,1^m1· · ·µ_k,i^mi,

with conventionαk,0=1. We haveµk+1,i=µk,iz^t_k+1^k+1Qk,i andαk+1,i=αk,iz^t_k+1^k+1Bk,i for all 1≤i≤k, andαk+1,k+1=αk+1,k.

Lemma 2. Let z₀=0and s₀=1. For all k=0, . . . ,g, we have the formula

π_k(x,y) = µk,0x^ek,

k i=0

∑

αk,ix^Bk,i z^s_iⁱ+c_i µk,ix^Qk,i

+αk,kx^Bk,ky

! .

Proof. This is correct fork=0: the formula becomesπ₀(x,y) = (x,y+c₀(x)). For k>0, we conclude by induction, using the recursive relations forB_k,i,µ_k,i andα_k,i above with the definitionπ_k(x,y) =πk−1(z^t_k^kx^qk,x^mk(z^s_k^k+c_k(x) +y)).

Given α an element of a ring L, we denote byα^1/e the residue class ofZ in L[Z]/(Z^e−α). For allk=0, . . . ,g, we define the ring extension

Lk:=Kk

z

1 e

1, . . . ,z

1 e

k

.

Note thatL0=K. Moreover, sincez^1/e_k has degreee`_k>1 overLk−1, the natural inclusionLk−1⊂Lkisstrict.

Remark 5. Note thatθ_k:=µ_k,0^−1/ek is a well defined invertible element of Lk (use Remark4), which by Lemma2plays an important role in the connections between pseudo-RPE and Puiseux series (proof of Proposition 1 below). In fact, we could replaceLkby the subringKk[θ_k]of sharp degreee_kf_k overK, see [24]. We useLk

for convenience, especially sincez^1/q_k ^k might not lie in Kk[θ_k]. The key points are:

θ_k∈Lkand the inclusionLk−1⊂Lkis strict.

Proposition 1. Let F∈K[[x]][y]be Weierstrass and considerS˜:=S(µ^−1/eT), where (µT^e,S(T)):=π_g(T,0). We have

S(T˜ ) =

∑

B>0

aBT^B∈Lg[[T]],

wheregcd(B₀, . . . ,B_k)|B and a_B∈Lkfor all B<B_k+1(with convention B_g+1:= +∞).

Moreover, we have for all1≤k≤g

a_Bk=





 ε_kz

1 qk

k if q_k>1 εkz_k+ρk if q_k=1

(15)

for someεk∈L^×_k−1andρk∈Lk−1. In particular a_Bk∈Lk\Lk−1.

Proof. Note first thatµ=µg,0 by Lemma2, so thatθg:=µ^−1/e is a well defined invertible element ofLg(Remark5). In particular, ˜S∈Lg[[T]]as required. Lemma2 applied at rankk=ggives

S(T) =

g

∑

k=0

αg,kT^Bk z^s_k^k+c_k µg,kT^eˆk

. (16)

Denote byθk:=µ_k,0^−1/ek ∈L^×_k (Remark5). Using the definitions ofµg,kandαg,k, a straightforward computation gives

µ_g,kθ_g^eˆk=θ_k∈Lk and α_g,kθ_g^Bk=

k

∏

j=1

µg,jθ_g^eˆj

mj

=

k

∏

j=1

θ_j∈Lk. (17) Combining (16) and (17), we deduce that ˜S(T) =S(θ_gT)may be written as

S(T˜ ) =

g k=0

∑

C_k(θ_kT^eˆk)T^Bk, C_k(T):= z^s_k^k+c_k(T)

k

∏

θj∈Lk[[T]]. (18) As ˆe_k=gcd(B₀, . . . ,B_k)divides both ˆe_iandB_ifor alli≤k, this forces gcd(B₀, . . . ,B_k) to divideBfor allB<B_k+1. In the same way, asLi⊂Lkfor alli≤k, we geta_B∈Lk

for allB<B_k+1. There remains to show (15). Asc_k(0) =0, we deduce that C_k(0) =z^s_k^k

k

∏

θ^m_j ^j =ε_kz^1/q_k ^k with ε_k:=

k−1

∏

θjz

−t j mk q j···qk

j ∈Lk−1, (19)

the second equality using the B´ezout relations_kq_k−t_km_k=1. Note thatε_k∈L^×k−1

(Remarks4and5). Letρ_kbe the sum of the contributions of the termsT^BiC_i(θ_iT^eˆi), i6=kto the coefficientT^Bk of ˜S. Soa_Bk=C_k(0) +ρ_k. AsB₁≤ · · · ≤B_gandk≥1, we deduce that ifC_i(θ_iT^eˆi)T^Bi contributes toT^Bk, theni<kso thatC_i(θ_iT^eˆi)T^Bi∈ Lk−1[[T^eˆk−1]]. We deduce thatρ_k∈Lk−1. Moreover,ρ_k6=0 forces ˆek−1|B_k. Sincem_k is coprime toq_k, we deduce from (9) thatq_k=1.

Remark6. In contrast to the Newton-Puiseux type algorithms of [22] which compute

∑Ba_BT^B(up to some truncation bound), algorithmPseudo-Irreducibleof Section 5.2only allows to compute(a_Bk−ρ_k)T^Bk,k=0, . . . ,gin terms of the edge data thanks to (15) and (19). As shown in this section, this is precisely the minimal information required to test pseudo-irreducibility and compute the equisingularity type. For in- stance, the Puiseux series ofF = (y−x−x²)²−2x⁴areS₁=T+T²(1−√

2)and S₂=T+T²(1+√

2)and we only compute here the “separating” terms−√

2T²and

√

2T². Computing all terms of the singular part of the Puiseux series of a (pseudo)- irreducible polynomial in quasi-linear time remains an open challenge.

Let us denote byW⊂K^gthe zero locus of the polynomial system defined by the canonical liftings ofP₁, . . . ,P_ginK[Z₁, . . . ,Z_g]. Note that Card(W) = f. Givenζ = (ζ₁, . . . ,ζ_g)∈W, the choice of somee^th-rootsζ₁^1/e, . . . ,ζ_g^1/einKinduces a natural evaluation map

ev_ζ :Lg'K z

1 e

1, . . . ,z

1

ge

−→K

and we denote for shorta(ζ)∈Kthe evaluation ofa∈Lgatζ. There is no loss to assume that whenζ,ζ⁰∈W satisfyζk=ζ_k⁰, we chooseζ_k^1/e=ζ_k^01/e. We thus have

(ζ₁, . . . ,ζ_k) = (ζ₁⁰, . . . ,ζ_k⁰) =⇒ a(ζ) =a(ζ⁰) ∀a∈Lk. (20) The following lemma is crucial for our purpose.

Lemma 3. Let us fixωsuch thatω^e=1and letζ,ζ⁰∈W . For all k=0, . . . ,g, the following assertions are equivalent:

1. a_B(ζ) =a_B(ζ⁰)ω^Bfor all B≤B_k. 2. a_B(ζ) =a_B(ζ⁰)ω^Bfor all B<B_k+1. 3. (ζ₁, . . . ,ζ_k) = (ζ₁⁰, . . . ,ζ_k⁰)andω^eˆk=1.

Proof. By Proposition 1, we havea_B∈Lk and ˆe_k|Bfor allB<B_k+1 from which we deduce 3)⇒2)thanks to hypothesis (20). As 2)⇒1)is obvious, we need to show 1)⇒3). We show it by induction onk. Ifk=0, the claim follows immediately since ˆe0=e. Suppose 1)⇒3)holds true at rankk−1 for somek≥1. IfaB(ζ) = a_B(ζ⁰)ω^B for allB≤B_k, then this holds true for all B≤B_k−1. Asε_k∈L^×k−1 and ρk∈Lk−1, the induction hypothesis combined with (20) gives εk(ζ) =εk(ζ⁰)6=0 andρk(ζ) =ρk(ζ⁰). We use now the assumptiona_Bk(ζ) =a_Bk(ζ⁰)ω^Bk. Two cases occur:

– Ifq_k>1, we deduce from (15) thatζ_k^1/qk=ζ_k^01/qkω^Bk, so thatζk=ζ_k⁰ω^qkBk. As ˆ

ek−1dividesq_kB_kandω^eˆk−1 =1 by induction hypothesis, we deduceζ_k=ζ_k⁰, as required. Moreover, we geta_Bk(ζ_k) =a_Bk(ζ_k⁰)thanks to (20), so thatω^Bk=1.

– Ifq_k=1, we deduce from (15) that ζ_k+ρ_k(ζ) =ω^Bk(ζ_k⁰+ρ_k(ζ⁰)). Asq_k=1 implies ˆek−1=eˆ_k|B_k, we deduce againω^Bk=1 andζ_k=ζ_k⁰.

AsB_k=∑s≤kmseˆs, induction hypothesis gives(ω^eˆk)^mk =1. Sincem_kis coprime to q_kand(ω^eˆk)^qk=ω^eˆk−1=1, this forcesω^eˆk=1.

Finally, we can recover all the Puiseux series of a pseudo-irreducible polynomial from the parametrisationπg(T,0), as required. More precisely :

Corollary 2. Suppose that F is pseudo-irreducible and Weierstrass. Then F admits exactly f distinct monic irreducible factors F_ζ ∈K[[x]][y]indexed byζ ∈W . Each factor F_ζ has degree e and defines a branch with classical Puiseux parametrisations (T^e,S˜_ζ(T))where

S˜_ζ(T) =

∑

B

a_B(ζ)T^B. (21)

The e Puiseux series of F_ζ are given byS˜_ζ(ωx^1e)whereω runs over the e^th-roots of unity and this set of Puiseux series does not depend of the choice of the e^th-roots ζ₁^1/e, . . . ,ζ^1/e.

Proof. AsF is pseudo-irreducible,H_g=y(see Section3.2) andπ_g^∗F(x,0) =0 by (13). We deduceF(T^e,S˜_ζ(T)) =0 for allζ ∈W. By (15), we havea_Bk(ζ)6=0 for allksuch thatq_k>1. Since gcd(B₀=e,B_k|q_k>1)) =gcd(B₀, . . . ,B_g) =eˆ_g=1, the parametrisation(T^e,S˜_ζ(T))is primitive, that is the greatest common divisor of the

exponents of the seriesT^eand ˜S_ζ(T)equals one. Hence, this parametrisation defines a branchF_ζ =0, whereF_ζ ∈K[[x]][y]is an irreducible monic factor ofFof degreee.

Thanks to Lemma3, these f branches are distinct whenζruns overW. As deg(F) = e f, we obtain all the irreducible factors ofFin such a way. Considering other choices of thee^throots of theζ_k’s would lead to the same conclusion by construction, and the last claim follows straightforwardly.

4.3 Pseudo-irreducible implies balanced

Proposition 2. Let F∈K[[x]][y]be pseudo-irreducible. Then each branch F_ζ of F has characteristic exponents(B₀;B_k|q_k>1),k=1, . . . ,g).

Proof. Thanks to Corollary2, all polynomialsF_ζ have same first characteristic expo- nentB₀=e. We also showed in the proof of Corollary2thata_Bk(ζ)6=0 for allk≥1 such thatq_k>1. We conclude by Proposition1.

Proposition 3. Let F ∈K[[x]][y]be pseudo-irreducible with at least two branches F_ζ,F_ζ0. We have

(F_ζ,F_ζ⁰)₀=M_κ, κ:=min

k=1, . . . ,g|ζ_k6=ζ_k⁰ .

and this value is reached exactly fˆκ−1−fˆ_κ times whenζ⁰runs over the set W\ {ζ}.

Proof. Noether-Merle’s formula (3) combined with Proposition2gives (F_ζ,F_ζ0)₀=

∑

k≤K

(eˆk−1−eˆ_k)B_k+eˆKCont(F_ζ,F_ζ⁰) (22) withK=max{k|Cont(F_ζ,F_ζ0)≥B_k}. Note that theB_k’s which are not characteristic exponents do not appear in the first summand of formula (22) (q_k=1 implies ˆe_k−1−

ˆ

e_k=0). It is a classical fact that we can fix any rootyofFfor computing the contact order in formula (2) (see e.g. [6, Lemma 1.2.3]). Combined with Corollary2, we obtain the formula

Cont(F_ζ,F_ζ⁰) =max

ω^e=1 v_T S˜_ζ(T)−S˜_ζ⁰(ωT)

. (23)

We deduce from Lemma3that v_T S˜_ζ(T)−S˜_ζ⁰(ωT)

=Bκ¯, κ¯:=min

k=1, . . . ,g|ζk6=ζ_k⁰orω^eˆk6=1 . Asω=1 satisfiesω^eˆk=1 for allk, we deduce from the last equality that the maximal value in (23) is reached for ω =1 (it might be reached for other values ofω). It follows that Cont(F_ζ,F_ζ⁰) =Bκ withκ=min

k|ζ_k6=ζ_k⁰ . We thus haveK=κand (22) gives(F_ζ,F_ζ⁰)₀=∑^κ_k=1(ˆek−1−eˆ_k)B_k+eˆκBκ=Mκ, the last equality by (10). Let us fixζ. As said above, we may chooseω=1 in (23). We havevT(S˜_ζ(T)−S˜_ζ⁰(T)) = Bκ if and only ifζ_k⁰=ζ_k fork<κ andζκ6=ζ_κ⁰. This concludes, as the number of possible such values ofζ⁰is precisely ˆfκ−1−fˆ_κ.

IfFis pseudo-irreducible, then it is balanced and satisfies both items of Theorem 2thanks to Propositions2and3. There remains to show the converse.

4.4 Balanced implies pseudo-irreducible

We need to show thatN_g=1 ifF is balanced. We denote more simplyH:=H_g∈ Kg[[x]][y], andπ_g(T,0) = (µT^e,S(T)). We denoteH_ζ,S_ζ,µ_ζ the images ofH,S,µ after applying (coefficient wise) the evaluation mapev_ζ :Kg→K. In what follows, irreducible means absolutely irreducible.

Lemma 4. Suppose that F is balanced. Then all irreducible factors of all H_ζ,ζ ∈W have same degree.

Proof. Let ζ ∈W and let y_ζ be a root of H_ζ. AsH_ζ divides (π_g^∗F)_ζ by (13), we deduce from Lemma2(rememberB_gg=B_g) that

F(µ_ζx^e,S_ζ(x) +x^Bgy_ζ(x)) =0.

Hence,y₀(x):=S˜_ζ(x^1e) +µ⁻

Bg e

ζ x^Bge y_ζ(µ⁻

1 e

ζ x^1e)is a root ofFand we have moreover the equality

degK((x))(y₀) =edeg

K((x))(y_ζ), (24)

where we consider here the degrees ofy₀andy_ζ seen as algebraic elements over the fieldK((x)). AsF is balanced, all its irreducible factors - hence all its roots - have same degree. Combined with (24), this implies that all roots - hence all irreducible factors - of allH_ζ,ζ ∈W have same degree.

Corollary 3. Suppose F balanced and N_g>1. Then there exists some coprime pos- itive integers(q,m)and Q∈Kg[Z]monic with non zero constant term such that H has lower boundary polynomial

H(x,¯ y) =Q(y^q/x^m)x^mdeg(Q).

Proof. AsN_g>1, the Weierstrass polynomialH=H_gis not pseudo-degenerate and admits a lower slope(q,m)(we can not haveH_g=y^NgasFwould not be square-free).

Hence, its lower boundary polynomial may be written in a unique way

H(x,¯ y) =y^rQ(y˜ ^q/x^m)x^mdeg(Q)˜ (25) for some non constant monic polynomial ˜Q∈Kg[Z]with non zero constant term and some integerr≥0. Ifr=0, we are done, takingQ=Q. Suppose˜ r>0. Letζ ∈W such that ˜Q_ζ(0)6=0. Applyingev_ζto (25), we deduce thatN(H_ζ)has a vertex of type (r,i), 0<r<d from which follows the well-known fact thatH_ζ =AB∈K[[x]][y], with deg(A) =rand deg(B) =qdeg(Q). By Lemma˜ 4, this forcesqto divider. Hence r=nqfor somen∈Nand the claim follows by takingQ(Z) =ZⁿQ(Z).˜

Lemma 5. Suppose F balanced and N_g>1. We keep the notations of Corollary3.

Let G be an irreducible factor of F inK[[x]][y]. Then e q divides n:=deg(G)and there exists a uniqueζ∈W and a unique rootαof Q_ζsuch that G admits a parametrisation (Tⁿ,S_G(T)), where

S_G(T)≡S˜_ζ(T^ne) +α

1 qµ⁻

Bg e

ζ T^a modT^a+1, (26)