Polar germs, Jacobian ideal and analytic classification of irreducible plane curve singularities

We examine the relationship between the analytic type of an irreducible plane curve singularity with a single characteristic exponent and the germs of curve defined by the elements of its Jacobian ideal, in particular its polar germs.


Introduction
Let ξ : f = 0 be a reduced non-empty germ of analytic plane curve, defined in a neighbourhood of the origin of C 2 by a convergent series f ∈ C{x, y} with no multiple factors. Here the ideal J(ξ ) = ( f x , f y , f ), f x and f y the derivatives of f , will be called the Jacobian ideal of ξ ; it does not depend on the series f defining ξ . The system of germs of curve defined by J(ξ ), J (ξ ) = {ζ : g = 0, g ∈ J(ξ ) − {0}}, will be called the Jacobian system of ξ . The polar germs of ξ -often called just polars in the sequel-are the germs of curve defined by equations h 1 f x + h 2 f y + h f = 0 with h 1 , h 2 , h ∈ C{x, y} and h i (0, 0) = 0 for at least one i = 1, 2. Generic germs (in the sense of [6], 2.7) of J (ξ ) are thus polar germs. In the sequel we will call generalized polars the elements of J (ξ ). The weighted cluster of base points of the Jacobian ideal, B P(J(ξ )), consists of the infinitely near points and multiplicities shared by generic polars ( [6], 7.2.13, 7.2.15); we will often refer to it as the (weighted cluster of) base points of the polars of ξ . Generic polars have no singular points outside B P(J(ξ )); therefore they all have the same topological type, which is determined by B P(J(ξ )); it is called the topological type of generic polars. If a local analytic automorphism ϕ maps ξ to ξ , then it maps B P(J(ξ )) onto B P(J(ξ )) preserving multiplicities; in particular, the topological type of generic polars is an analytic invariant of the germ.
Since the nineteenth century, polar curves and polar germs are known to enclose deep information: in a global context, they determine the singular points of a reduced projective plane curve and may be used to control the resolution of its singularities ( [10], IV.II.14, for instance); in the local case, the base points of the polars of a reduced germ of curve ξ are known to determine the topological type of ξ ( [6] 8.6.4, [2] 2.4); in a higher-dimensional context, polar varieties provide the so called polar invariants ( [11,13,18]).
Interest in polar germs was increased after an example due to Pham ( [17]), showing that the topological type of generic polars of a germ of curve ξ depends on the analytic type of ξ , and not only on its topological type. As a consequence, rather evident characters of the polar germs of ξ may be used to unveil much more hidden analytic characters of ξ : for instance, topologically equivalent germs of curve may be shown not to be analytically equivalent by showing that their generic polars are not topologically equivalent. For irreducible germs general enough, the base points of the polar germs provide a number of continuous analytic invariants ( [5]). Regarding the whole Jacobian ideal, Mather and Yau proved in [14] that the quotient algebra C{x, y}/J(ξ ) determines the analytic type of ξ . More recently, a deep result first claimed by Hefez and Hernándes [12], characterizes the analytically relevant coefficients of the Puiseux series of an irreducible germ ξ in terms of the intersection multiplicities of ξ and the germs in its Jacobian system J (ξ ).
The main purpose of this paper is to go deeper into the relationship between polars -and generalized polars-and analytic classification of irreducible germs of plane curve with single characteristic exponent m/n, on the basis of the analytic information recently given in [7] and examining in particular how much of this information is retained by the polars.
Let us first recall some results from [7]. For fixed coprime integers m, n, m > n > 1, are considered there the germs of curve γ with equations which represent all analytic classes of irreducible germs with single characteristic exponent m/n. As usual, the point (i, j) ∈ R 2 is associated to the coefficient A i, j . Then -with the exceptions noted below -are determined there the coefficients A i, j whose variation changes the analytic type of γ (relevant coefficients) and, among them, those whose value is -but for a finite conjugation-determined by the analytic type of γ (continuous invariants). The excepted coefficients are those corresponding to the integral points in the interior of a certain triangle T : each of them may be either non-relevant or a continuous invariant, depending on the values of preceding -by the ordering induced by the value of ni + m j-coefficients. The coefficients corresponding to the points in T are called conditional invariants. For the easiest representative example, see Fig. 1. Now, for the main results obtained here, those relative to polars are essentially negative, showing that the topological type of generic polars and the base points of the polars retain only partial information on the analytic type of the germ. More precisely, for germs γ with single characteristic exponent m/n as above, we obtain: Generalized polars enclose much more information, as shown in the already cited [14] and [12]. Regarding them, here we obtain: -A proof of a single characteristic exponent version of Hefez and Fernandes' characterization: a coefficient A i, j , ni + m j > nm + σ (γ ), is relevant if and only if there is no generalized polar ζ with [ζ · γ ] = ni + m j (7.5). It results a rather explicit determination of analytic automorphisms causing the variation of each non-relevant coefficient and leaving the preceding coefficients invariant (7.7). -The existence of a continuous broken line -the staircase line of γ -that separates the points in T corresponding to conditional invariants into those corresponding to continuous invariants and those corresponding to irrelevant coefficients, see The content is organized as follows: Sects. 3 and 4 are devoted to polars; besides the results quoted above, they contain a close examination of Newton polygons of polars and the way they are shaped by the values of certain continuous invariants. Sections 5 and 6 are preparatory; in the second one there is a new way to obtain coordinates on infinitesimal neighbourhoods which may be of independent interest. Section 7 deals with Hefez and Fernandes' characterization and the staircase line. Section 8 completes the information about the intersection multiplicities [ζ · γ ], ζ a generalized polar; this provides a characterization of the Zariski invariant (8.2). In Sect. 9, a couple of easy tools are adapted from computational algebra and applied to compute the Zariski invariant (9.3) and the staircase line (9.9). They are used in three examples: in 9.4 a non-evident Zariski invariant is computed, 9.10 sheds light on an already known case of jumping Tjurina number and 9.11 describes the different possibilities of relevance of coefficients for germs with characteristic exponent 17/6 and minimal Zariski invariant.

Preliminaries
We place ourselves under the general conventions of [6] and, more especifically, under those of [7]. In particular, germs of complex analytic plane curves will often be called just germs. Positive coprime integers m, n, m > n > 1 will be fixed throughout and we will mainly consider irreducible germs γ , defined in a neighbourhood of the origin O of C 2 by a convergent power series of the form (1) as it is well known ( [7], Section 2, for instance), they are irreducible, have single characteristic exponent m/n, and a Puiseux series Furthermore, any irreducible germ with single characteristic exponent m/n is analytically equivalent to a germ γ as before. Therefore, our analytically invariant conclusions will hold for arbitrary irreducible germs with a single characteristic exponent. The multiplicity of a point O on a curve or germ of curve ξ will be denoted e O (ξ ). The Newton polygon of a germ ξ : g = 0, g = i, j≥0 B i, j x i y j will be taken to be the union of the compact sides and vertices of the border of the convex envelope of {(i, j) ∈ R 2 |B i, j = 0} + (R + ) 2 . Newton polygons will be drawn on a plane N = R 2 -the Newton plane-in which we conventionally assume that the first axis is horizontal, oriented from left to right, while the second axis is vertical, oriented from bottom to top. We will take the Newton polygons and their sides -all with negative slope-oriented from top left to bottom right. N 2 will denote the set of points of N with non-negative integral coordinates. Non-zero vectors v on N with both components non-negative will be called positive vectors, indicated v > 0.
We will take ni + m j as the twisted degree of any non-zero monomial ax i y j . The twisted degree td (resp. twisted order to) of a non-zero polynomial (resp. series) g will be the highest (resp. lowest) twisted degree of the non-zero monomials of g, and we will take to(0) = ∞. The twisted initial form of g ∈ C{x, y} − {0} is the sum of all its monomials of minimal twisted degree, usually denotedḡ in the sequel. The integer ni + m j will be also taken as being the twisted degree of the point (i, j), which represents ax i y j on the Newton plane.
As in [7], a point (i, j) ∈ N 2 with ni + m j > nm will be called standing if i < m − 1 and j < n − 1, and non-standing otherwise. The monomials corresponding to standing points and their coefficients will be called standing monomials and standing coefficients, respectively.
Remark 2.1. The reader may easily check that if p is a standing point, then no point p ∈ N 2 , p = p, has td( p) = td( p ). In particular, if g ∈ C{x, y} has to(g) = td( p), p a standing point, then the twisted initial form of g has a single monomial, and such a monomial corresponds to p. It easily turns out that the set of twisted degrees of the standing points and the set of twisted degrees of the non-standing points are disjoint, and also that the latter is where n, m is the semigroup generated by n, m.
A coefficient A h,k of the equation (1) of a germ γ , say with nh + mk = d, is said to be (analytically) irrelevant if and only if for any α ∈ C there is a germ γ α , analytically equivalent to γ and defined by an equation the dots meaning terms of higher twisted degree. Irrelevant coefficients may be turned into zero by the action of a suitable local automorphism, with no modification to the other coefficients of equal or smaller twisted degree. This is called eliminating the coefficient, see [7], Section 5 for details. All non-standing coefficients are irrelevant ( [7], 6.1). Relevant coefficients are of course those which are not irrelevant.
The reader is referred to [7], Sect. 1, for a description of which coefficients in (1) are relevant. In particular, the coefficients whose variation causes a variation of the analytic type with at most finitely many occurrences of each analytic type, are called continuous invariants. For a precise definition, a coefficient A h,k , again with nh + mk = d > nm, in (1) is an continuous invariant of the germ γ : f = 0 if and only if there is a non-constant polynomial φ ∈ C[X ] such that no germs γ : y n − x m + nm<ni+m j≤d A i, j x i y j + ρ x h y k + · · · = 0, = 1, 2, are analytically equivalent if φ(ρ 1 ) = φ(ρ 2 ), see [7], Sect. 9. Obviously, continuous invariants are relevant. Coefficients which may or may not be continuous invariants, depending on the values of lower twisted degree coefficients, are called conditional invariants.
For the convenience of the reader, we sketch below some elementary facts relating Newton polygons and infinitely near points that will be used in the sequel. The origin O of C 2 , the points infinitely near to O on the x-axis and the satellite points of the latter will be called initial points. If ζ is an irreducible germ with origin at O and Puiseux series S = cx s/r + . . . , c = 0 , r = e O (ζ ), we will call cx s/r the initial term, and s/r the initial exponent, of ζ . The next lemma contains well known facts, see for instance [6], 5.3.1.

Lemma 2.2.
The hypothesis and notations being as above, the initial points ζ is going through, and their multiplicities on ζ , depend only on the pair (s, r ) and not on ζ itself: the quotients appearing in the Euclidean algorithm for gcd(s, r ) determine the points themselves, while the remainders are the multiplicities of the points. Also, irreducible germs with different initial exponents share no points other than initial points.
Assume that ξ : i, j≥0 α i, j x i y j = 0 is a germ of curve with origin at O and not containing any of the two germs of the coordinate axes. Let us recall (see for instance [6], 2.2) that each branch of ξ comes associated with a side of the Newton polygon N(ξ ) of ξ . If is a side of N(ξ ) which has width s, height r and lowest end i 0 , j 0 , then the branches associated with are ζ k , k = 1, . . . , h, h > 0, where each ζ k has initial term c k x s k /r k , r k = e O (ζ k ), s k /r k = s/r and c k is a solution of (i, j)∈ α i, j Z j− j 0 = 0 -the equation associated to . Also, if the branches are repeated according to their multiplicities as irreducible components of ξ , k s k = s and k r k = r . Lemma 2.3. Let ξ be a germ not containing the germ of the y-axis. Then the initial points ξ goes through, as well as their multiplicities on ξ , are determined by N(ξ ).
Proof. If the germ of the x-axis is a branch of ξ , then it is a smooth branch and its multiplicity as an irreducible component of ξ is the second coordinate of the lowest vertex of N(ξ ). All the other branches of ξ correspond to sides of N(ξ ); let be one of these sides. The notations being as above, take ξ = k ζ k , where the ζ k are the branches of ξ associated with , repeated according to multiplicities. Being s/r = s k /r k for all k, the quotients in the Euclidean algorithms for gcd(s, r ) and each gcd(s k , r k ) are the same; hence (2.2) ξ and any of the ζ k have the same initial points, and these are determined by (s, r ). Further, since s = k s k and r = k r k , the remainders in the Euclidean algorithm for gcd(s, r ) equal the sums of the corresponding remainders in the algorithms for the gcd(s k , r k ); using 2.2 again, the former are the multiplicities of the initial points on ξ . By adding up for the different sides of N(ξ ) and taking into account the germ of the x-axis if it is a branch of ξ , the claim follows.

Polars
The results in this section and the next one show that, for irreducible germs of curve with a single characteristic exponent, both the topological type of generic polars and the base points of the polars give only partial information on the analytic type of the germ.
If ξ : g = 0 is a reduced germ of plane curve, we will consider the pencil of polar germs generated by the polars ξ x : g x = 0 and ξ y : g y = 0. The pencil P g depends on the choices of the coordinates x, y and the series g. Nevertheless, by [6], 8.5.7, P g and J(ξ ) have the same cluster of base points and so, in particular (by [6], 7.2.1), all but finitely many polars in P g have the topological type of generic polars. In the sequel, the base points of the polars of ξ and the topological type of generic polars of ξ will be obtained from a pencil P g . To this end we will often use the Newton polygon shared by all but finitely many members of P g ; the series g being fixed, it will be noted PN(ξ ).
Let γ be a germ defined by an equation as (1); as said, it is irreducible and has single characteristic exponent m/n. The easiest analytic -non-topologicalinvariant of γ is the Zariski invariant σ = σ (γ ); it may be defined as and ni + m j < nm + σ have been previously turned into zero by the action of a suitable analytic automorphism ( [7], 7.14). We take min ∅ = ∞; the germs with σ = ∞ (quasihomogeneous germs) are all analytically equivalent to y n − x m = 0, and therefore have little interest regarding the analytic classification. If σ = ∞, then the only point ω = (ω 1 , ω 2 ) ∈ N 2 with twisted degree nm + σ is called the Zariski point of γ . When finite, the Zariski invariant obviously carries the same information as the Zariski point. The next example shows that the Zariski invariant is not determined by the topological type of generic polars: The variation of the topological type of generic polars of the same germs is also worth some attention: Example 3.2. Take the same germs γ α as in Example 3.1 above, this time with α = 0 in order to have the same Zariski point (5, 5) for all. As seen in 3.1, the topological type of generic polars is constant for α 3 = −1323/500. However, it changes for α 3 = −1323/500, because then the equation associated to the only side of PN(γ α ) acquires a double root. Therefore, also the analytic type of γ α changes for α 3 = −1323/500. This change is, however, just part of a continuous variation of the analytic type of γ α with α 3 which otherwise does not affect the topological type of generic polars.
Indeed, by [7], 11.1, the coefficient corresponding to (15, 1) is a continuous invariant of γ α . For a correct reading of it, the equation needs to be normalized by turning the coefficient of the term corresponding to the Zariski point into 1. For each fixed α = 0, this is achieved by replacing γ α with its inverse image by the local automorphism x * = α 7 x, y * = α 17 y, which is and gives 1/α 3 as the value of the invariant.
Back to considering an arbitrary germ γ with equation (1), we will pay some attention to the Newton polygons of polar germs of γ , and in particular to PN(γ ). As it is well known, the Newton polygon of a germ of plane curve depends on the relative position of the germ and the coordinate axes. Therefore PN(γ ) cannot be expected to be an analytic invariant of γ , and in fact it is not, see Example 3.10 below.
For generic germs γ (that is, for all γ with equation (1) but those whose coefficients A i, j satisfy certain finitely many polynomial equations), the Newton polygon PN(γ ) was determined in [4]. Nevertheless, examining the relationship between the polars and the analytic type of γ requires considering all germs γ , as we will do next.
If g ∈ C{x, y}, the set of points of N corresponding to the non-zero coefficients of g -the Newton diagram of g-will be denoted (g) in the sequel. First of all note that the derivation ∂/∂y (resp. ∂/∂ x) acts on the points of ( f ) by deleting the points on the first (resp. second) axis and shifting the other points one unit downwards (resp. leftwards). Therefore, no matter what the values of the coefficients A i, j are, ( f y ) always has (0, n − 1) as its lowest point on the second axis, while ( f x ) contains (m − 1, 0) and, being m/n > 1, no point (0, r ), r ≤ n − 1. It follows: As a consequence, the topological type of generic polars of γ ceases giving analytic information on γ if the Zariski invariant of γ is m − n or higher: Proposition 3.5. Generic polars of irreducible germs of curve with single characteristic exponent m/n and Zariski invariant σ ≥ m−n have all the same topological type: if d = gcd(n −1, m −1), n −1 = r d and m −1 = sd, then they are composed of branches ξ k , k = 1, . . . , h, each ξ k with a Puiseux series S k = α k x s/r + · · · , α k = 0 and α r k = α r for k = .
Proof. By [7] 6.3, up to replacing γ with an analytically equivalent germ, we may assume that γ is given by an equation as (1) in which all non-standing coefficients A i, j (i.e., those with either i ≥ m − 1 or j ≥ n − 1) are zero. After this, since in particular A m−1,1 = 0, both ends of belong to the Newton diagram of λ( f x ) + μ( f y ) = 0 for λ, μ = 0. Furthermore, by [7], 7.14, if A i, j = 0, then ni + m j ≥ nm +σ ≥ nm +m −n. The monomials produced by deriving a non-zero monomial A i, j x i y j correspond to the points (i − 1, j) and (i, j − 1), and have twisted degrees Since the maximum of the twisted degrees of the points on is nm − n, reached at the end (m − 1, 0) only, and neither of the above points is (m − 1, 0) (because A m−1,1 = 0), they both lie strictly above . This proves that is the only side of PN(γ ) and also that no point on other than its ends corresponds to a non-zero coefficient of the equation of a polar in P f . Then the equations associated to have no multiple roots and the claim follows by just computing the initial terms of the Puiseux series of the polars For an arbitrary γ , PN(γ ) is not far away from , see  Proof. the integers n, m being coprime, no point in N 2 other than (0, n − 1) lies on the line ni + m j = nm − m. Hence, by 3.6, the first side of PN(γ ) has slope strictly higher than −n/m, and the same holds for the other sides due to the convexity of the polygon.
The equations of the polars in P f other than f x = 0 may be written in the form θ f x + f y , θ ∈ C; then, some of their coefficients remain constant: Lemma 3.8. The coefficients of h θ = θ f x + f y corresponding to the points in other than (m − 1, 0), are all independent of θ .
Proof. As noticed in the proof of 3.6, the Newton diagram of f x lies in the half-plane ni + m j ≥ nm − n, whose only common point with is (m − 1, 0).
Letˆ be the image of by the translation of vector (0, 1). By 3.8, the coefficients of h θ corresponding to the points Proof. The Zariski point ω = (ω 1 , ω 2 ) being a standing point, it holds n − 2 ≥ ω 2 ≥ 1, and hence, using that nω 1 + mω 2 > nm, also ω 1 > 2m/n . Therefore, according to their definition (see [7], Section 12), all points (i, j) corresponding to conditional invariants satisfy either Since the maximum of the twisted degrees of the points inˆ is nm + m − n, the claim follows.
There may be non-standing points in the interior ofˆ ; the elimination of their corresponding -necessarily irrelevant-coefficients by local automorphisms may change PN(γ ), but of course not the topological type of generic polars. Here is an example: 10. Take γ : f = y 4 −x 21 +4x 6 y 3 = 0. The point (6, 3) is non-standing and belongs to both ( f ) andˆ . The Newton polygon PN(γ ) has vertices (3, 0), (6,2) and (20, 0), the intermediate one being originated by (6, 3) by derivation. The inverse image of γ by the local automorphism x * = x, y * = y−x 6 is γ * : y 4 −x 21 − 6x 12 y 2 + 8x 18 y − 3x 24 = 0, which has not the irrelevant monomial corresponding to (6,3). The Newton polygon PN(γ * ) has a single side, with vertices (3, 0) and (18,0). Needless to say, see Sect. 1, generic polars of γ and γ * have the same topological type, which may be directly checked by the reader in this case. The difference between PN(γ ) and PN(γ * ) is due to the different positions of the polars with respect to the coordinate axes.
To avoid the effect of non-standing coefficients, assume that those with twisted degree less than 2nm − n − m + 1 (or all) have been turned into zero by replacing γ with an analytically equivalent germ, still named γ ( [7], 6.2 and 6.3). Then, by [7] 7.14, all critical coefficients are zero, but the one corresponding to the Zariski point ω = (ω 1 , ω 2 ). By 3.9 and the previous elimination of non-standing coefficients, the points inˆ corresponding to a non zero A i, j are ω, in case it belongs

Base points of polars
We will write B P(J(γ )) = PBP if no confusion may arise. Fix ξ : h = λ f x +μf y = 0 to be a polar with Newton polygon PN(γ ) and going sharply through PBP. We take h = f x if A 1,n = 0 and h = f x + f otherwise, in order to have (0, n) ∈ (h ) in all cases. Proof. The first claim is clear after the definition of h . The second one follows from it, as we already know (proof of 3.6) that the Newton diagram of f x is contained in the half-plane ni + m j ≥ nm − n. The third claim follows then from an easy argument similar to the one used in the proof of 3.7.

Lemma 4.2.
Any monomial x i y j , with ni + (m − 1) j ≥ nm − n defines a germ of curve going through PBP.
Proof. The polar ξ : h = 0 obviously goes through PBP. By 3.7 and 4.1, no branch of ξ has the same initial exponent as a branch of ξ . As a consequence the points shared by ξ and ξ are all initial points (by 2.2). Then, since all points in PBP belong to ξ , all the points in PBP the polar ξ is effectively going through, are initial points. For any non-initial point q ∈ PBP, ξ does not effectively go through q and so, the multiplicity at q of the virtual transform with origin at q remains the same if ξ is replaced with any germ ξ having the same effective multiplicities as ξ at the initial points. As a consequence any such ξ also goes through PBP (see [6], 4.1, 4.2). By 2.3, this occurs if N(ξ ) = N(ξ ).
By the hypothesis and 4.1, the point p = (i, j) does not lie below N(h ); then, for all values of α but at most one, in particular for a certain non-zero α. By the above, the germ ξ : h = 0 goes through PBP, after which so does the germ defined by then the weighted cluster of base points of the polars of γ and the topological type of its generic polars remain the same if any of the coefficients A i, j , ni +m j ≥ nm +m is allowed to take arbitrary values. Some of the above coefficients A i, j , ni + m j ≥ nm + m may be continuous invariants; then their values cannot be read from PBP(γ ). This is the case for m/n = 13/6, see [7], Example 12.18.

Proof of 4.3:.
Of course, only the part of the claim regarding PBP needs to be proved; the other has been included for the sake of completeness. We will prove that the germ obtained from γ by an arbitrary modification of A i, j still has two polars going sharply through PBP and sharing no points outside of it; from this, the claim directly follows.
Fix i, j satisfying the inequality of the claim and for any α ∈ C let γ α be the germ γ α : f + αx i y j = 0. For each polar of γ , ξ : g = λ f x + μf y = 0, going sharply through PBP and with Newton polygon PN(γ ), we will prove next that the polar of γ α is also going sharply through PBP.
Note first that all branches of ξ are tangent to the x axis, because their initial exponents are higher than m/n (by 3.7). Also, all the infinitely near points shared by ξ and the x-axis belong to PBP, because these points depend only on the Newton polygon the germ (by 2.3), and therefore are also shared by any other polar in P f going sharply through PBP and with Newton polygon PN(γ ).
The coefficients λ, μ still being fixed, the polars ξ α , together with the germ ξ ∞ : iλx i−1 y j + jμx i y j−1 = 0, describe a pencil of germs of curve B, generated by ξ 0 = ξ and ξ ∞ . The germ ξ obviously goes through PBP, and, after direct computation, so does ξ ∞ due to the hypothesis ni + m j ≥ nm + m and 4.2. All germs in B are thus going through PBP, and in particular so do the polars ξ α .
Select a branch ζ of ξ . By 3.7, it has a Puiseux parameterization x = t r , y = ct s + · · · , with s/r > m/n. Then the only non-zero monomial of minimal (r, s)-twisted degree in f x is −mx m−1 and hence, after substitution, the intersection multiplicity of ζ and the polarξ : . In particular we chose such aξ to be going sharply through PBP. Beingξ = ξ , ζ andξ share no point outside PBP, otherwise such a point would be a base point. Using then the Noether formula ( where ν p is the virtual multiplicity of p in PBP and e p (ζ ) its (effective) multiplicity on ζ .
By using again the above Puiseux parameterization and the hypothesis, we obtain: Let q be the first point on ζ not in PBP. Since ξ goes sharply through PBP, q and all points following it on ζ are free and have multiplicity one on ζ . The point q clearly does not belong to ξ ∞ : x i−1 (iλy + jμx)y j−1 = 0 because, as noted at the beginning of the proof, ζ is tangent to the x axis and all points shared by ξ and the x-axis are base points. Nevertheles, iterated use of the virtual Noether formula ( [6], 4.1.2) and the inequality (3) show that the virtual transform with origin at q, ξ ∞ , of ξ ∞ (relative to the virtual multiplicities in PBP) is non empty, and therefore equal to k E, where E is the germ of the exceptional divisor at q (a smooth germ because q is free) and k a positive integer.
The virtual transforms with origin at q of the germs in B describe a pencilB generated byξ ∞ and the virtual transformξ of ξ . The latter is in fact the strict transform of ξ -because ξ goes sharply through PBPand therefore is a smooth germ transverse to E. Using suitable equations z 1 ofξ and z 2 of E as local coordinates at q, the virtual transformξ α of ξ α has equation z 1 + αz k 2 = 0, k a positive integer, and therefore is a smooth germ transverse to E. In particularξ α has not E as a component and therefore is also the strict transform of ξ α , which has then q as a non-singular point. Call ζ α the only branch of ξ α containing q: ζ and ζ α , as any two irreducible germs sharing a non-singular point q, have the same points preceding q and the same multiplicities at them.
We have thus associated to each branch ζ of ξ , a branch ζ α of ξ α in such a way that both ζ and ζ α have the same effective multiplicities at the points in PBP and the same first point outside PBP; such a point is non-singular for both branches and does not belong to other branches of ξ . Since, clearly from its definition, ξ α has the same multiplicity n − 1 as ξ , ξ α has no branch other than those associated to the branches of ξ . In particular ξ α has no singular points outside PBP. It suffices to add up the multiplicities of their branches at each point of PBP to see that ξ α and ξ have the same multiplicities at the points of PBP and therefore that ξ α goes sharply through PBP. Now, to close, take λ , μ , λ /μ = λ/μ, such that still the polar of γ , ξ : g = λ f x +μ f y = 0 goes sharply through PBP and has Newton polygon PN(γ ). Then, the above arguments applying also to ξ , the corresponding polar of γ α , ξ α , goes sharply through PBP too.
As it is well known ( [6], 6.4, for instance), the intersection number of two different polars of a reduced germ of curve is the Milnor number of the germ, which in our case, for both γ and γ α , equals nm − n − m + 1. Therefore, if still ν p denotes the virtual multiplicity of p in PBP, the last equality using the Noether's formula ( [6], 3.3.1) and the fact that both ξ and ξ go sharply through PBP and share no points outside it. Since also ξ α and ξ α go sharply through PBP, the equality displayed above shows that they share no point outside PBP. Therefore the weighted cluster of base points of the polars of γ α is PBP, as claimed.

Intersecting with generalized polars
Still taking γ : in this section we will consider the finite intersection multiplicities of γ and its generalized polars, and prove that these intersection multiplicities may be replaced with twisted orders of equations of generalized polars, and conversely. We take In particular, is a semigroup. The positive integers that do not belong to will be called the gaps of J(γ ). The Jacobian ideal being intrinsically related to the germ, analytically equivalent germs γ, γ have (γ ) = (γ ).
Remark 5.2. Both nm −m and nm −n belong to , because they are the intersection multiplicities of γ and the polars f y = 0 and f x = 0 (by either the Plücker formula ( [6], 6.3.2) or a direct substitution). If γ has finite Zariski invariant σ , then, nm + σ ∈ . Indeed, assuming, as allowed, all non-standing coefficients of f to be zero, Using the equality of differentials on γ , it is direct to check that a suitable shifting of is the set of the orders of the non-zero differentials on γ , which is the way in which appears in [12]. In our case, may be handled as the set of the twisted orders of the non-zero elements of the Jacobian ideal, which will be quite useful in the sequel: Proof. For an arbitrary germ of curve ζ : g = 0, assume to(g) = d and g = g d +. . . where g d is the twisted initial form of g and the dots represent terms of higher twisted degree. Substituting the Puiseux parameterization of γ , x = t n , y = t m + . . . , it is g d (1, 1) + . . . )), the dots meaning now terms of higher degree in t. Therefore we have [ζ ·γ ] ≥ to(g) and the equality holds if and only if g d (1, 1) = 0. Now assume that r ∈ . Then there is a germ ζ : g = 0, g ∈ J(γ ), for which r = [ζ · γ ]. By the above, d = to(g) ≤ r . If the equality holds, r is indeed the twisted order of a non-zero element of J(γ ). Otherwise the twisted initial form g d of g satisfies g(1, 1) = 0. An easy computation shows that then g d has a factor y n − x m , say g d = (y n − x m )Q. Substracting f Q cancells the twisted initial form of g, hence g = g − f Q ∈ J(γ ) has twisted order at least d +1; the germ it defines, ζ : g = 0, still has [ζ · γ ] = r and it is enough to use decreasing induction on r − d.
Assume d = to(g), g ∈ J(γ ) − {0}. Take ζ : g = 0. Again it is d ≤ [ζ · γ ] and, if the equality holds, d ∈ . Otherwise, the twisted initial form g d of g satisfies g(1, 1) = 0, and therefore, as above, has factor y n − x m . Assume g d = (y n − x m ) k H , where k > 0, all monomials of H have twisted degree d − knm and H (1, 1) = 0. The twisted initial form of y f y is ny n , after which it is clear that In particular g d + (ny n ) k H = 0 and so, it has twisted degree d and is the twisted initial form of g = g + (y f y ) k H ; then, obviously, g ∈ J (γ ) and has twisted order d. If ζ is the germ ζ : g = 0, then, due to the inequality (4), [ζ · γ ] = d, as wanted.
Remark 5.4. Since J(γ ) is an (x, y)-primary ideal, any g ∈ C{x, y} with td(g) high enough belongs to J(γ ): it follows thus from 5.3 that J(γ ) has finitely many gaps.
Remark 5.5. Lemma 5.3 is not true for irreducible germs with two or more characteristic exponents: if the characteristic exponents are 6/4, 9/4, then n = 4, m = 6 and, by the Plücker formula ( [6], 6.3.2), the intersection multiplicity with a generic polar is 21.

Coordinates and translations in first neighbourhoods
Let s be a positive integer. We will call q s the point on γ in the s-th neighbourhood of the last satellite point, and E s the first neighbourhood of q s . E s is a projective line and, q s being free, contains a single satellite pointq; E s −q, which is the set of free points in E s , may thus be taken as an affine line with improper (or ideal) pointq. In [7], Section 4, it is shown that coefficients of twisted degree nm + s of the equations of irreducible germs through q s may be taken as affine coordinates of their points in E s . In this section we will show a different way of getting affine coordinates on E s , and use them to control the action of local automorphisms of C 2 on E s . Let η : h = 0, h = y n − x m + . . . , the dots meaning terms of higher twisted degree, be an irreducible germ with single characteristic exponent m/n going through q s . Let x = t n , y = t m u(t), u(0) = 1, be a Puiseux parameterization of a second germ η , also irreducible, with single characteristic exponent m/n and going through q s . The germs η and η sharing the point q s , by the Noether formula, o t (h(t n , t m u(t)) = [η · η ] ≥ nm + s. and we have: There is an affine coordinate on E s such that, for any η as above, if h(t n , t m u(t)) = αt nm+s + . . . , then the point on η in E s has coordinate α.
Proof. Pick non-negative integers i, j with ni + m j = nm + s and take h a = h + ax i y j . By [7], 4.3, there is an affine coordinate on E s such that for any a ∈ C, a is the affine coordinate of the point on η a : h a = 0 in E s . By substituting the Puiseux parameterization of η in h a , h a (t n , t m u(t)) = h(t n , t m u(t)) + at nm+s u j = (α + a)t nm+s + . . . .
Again by the Noether formula, η and η a have the same point in E s if and only if [η · η a ] > nm + s, which, by the above equality, is equivalent to α = −a, hence the claim.
The coordinate of 6.1 depends on the choice of the series h defining η. It will be called the coordinate associated to h, or just the h-coordinate. The series h will always be taken of the form h = y n − x m + . . . . Obviously the h-coordinate of the point on η is α = 0. A change between affine coordinates z,ẑ, of the form z = z + b, b ∈ C, will be called unimodular. We have:  = α + B(1, 1), which proves the claim.
If, in an affine line with a fixed affine coordinate z, a translation is given by the equation z * = z + b, we will call the complex number b the modulus of the translation. As it is clear, the modulus may change by changing the affine coordinate, but it remains the same if the change of coordinate is unimodular.

.2) induced by ϕ on E s .
Proof. Since ϕ leaves invariant q s , again by the Noether formula, [γ · ϕ * (γ )] ≥ nm + s and an equality as the one in the claim does hold. The automorphism ϕ being principal, the series ϕ * ( f ) still has the form ϕ * ( f ) = y n − x m + . . . and we may consider the coordinate associated to it. Using this coordinate, by 6.1, β(ϕ) is the coordinate of the point on γ in E s . This point is the image by ϕ of the point on ϕ * (γ ), which has coordinate 0; thus, still using the coordinate associated to ϕ * ( f ), the modulus of the translation is β(ϕ). The same holds using the coordinate relative to f due to 6.2.
In particular: Corollary 6.4. The mapping ϕ → β(ϕ) is a group homomorphism between the group of principal automorphisms at O leaving q s invariant and the additive group of C.

Analytic relevance and gaps of J(γ )
This section is devoted to give a proof of the theorem below, which sets a very interesting link between analytically relevant coefficients of a germ and gaps of its Jacobian ideal. It was first proved by Hefez and Hernandez in 2011. For the original version, which applies to germs with many characteristic exponents, the reader is referred to [12]. As an application, we will also prove the existence of a continuous broken line -the staircase line of the germ-that separates the points representing conditional invariants into points representing continuous invariants and points representing irrelevant coefficients.

Theorem 7.1. Let γ be an irreducible analytic germ of plane curve with a single characteristic exponent m/n, finite Zariski invariant σ and equation
Then, for any integer s > σ , a coefficient A i, j , with twisted degree ni +m j = nm+s, is analytically relevant if and only if there is no generalized polar ζ : Remark 7.2. The theorem does not cover the coefficients A i, j with twisted degree nm + s, s ≤ σ , but the relevance of these is clear after [7], Section 7: if the inequality is strict, they are either non-standing coefficients -and therefore analytically irrelevant-or critical -hence relevant-coefficients taking their critical values; the only coefficient A i, j with twisted degree nm + σ is critical and therefore relevant; it is the critical coefficient of highest twisted degree ( [7], 9.3). Therefore, when the theorem applies, the coefficients corresponding to gaps are all continuous invariants (see [7], Sect. 1).
It is worth noting that, in general, just because of 7.1, setting a certain positive integer r to be, or to be not, a gap of J(γ ) may impose constraints on coefficients of lesser twisted degree that are continuous invariants. For instance, back to Zariski's example m/n = 7/6 ( [19] V.5), as presented in [7] 12.15, all classes of germs with minimal Zariski invariant are represented by germs and A 4,3 and A 3,4 are continuous invariants. By the theorem, setting 52 as a gap is equivalent to the relevance of the coefficient A 4,4 , which in turn, by [7] 12.15, is equivalent to the equality 63A 2 4,3 − 56A 3,4 = 20. Therefore, setting the semigroup as a first invariant in order to analytically classify irreducible germs, makes clear which coefficients are continuous invariants, but leaves obscure which is the range of variation of each of them.

Proof of 7.1:.
For the only if part we will assume that there is a generalized polar ζ as in the claim and construct from it a local flow whose members act transitively on the first neighbourhood of the point in the s-th neighbourhood of the last satellite, thus proving the irrelevance of the coefficient. For the converse, assuming the coefficient irrelevant, we will use results from [7] to construct a particular family of local automorphisms, and from it a generalized polar as wanted.
We will make frequent use of the fact, already seen in the proof of 5.3, that any germ δ : h = 0 has [δ · γ ] ≥ to(h), the inequality being strict if and only if the twisted initial form of h has a factor y n − x m . The claim being invariant by local automorphisms at O, in the sequel we assume that the series f defining γ has the form where (ω 1 , ω 2 ) is the Zariski point and therefore nm + σ = nω 1 + mω 2 ( [7], 7.15). Fix s > σ, and assume that there is a generalized polar ζ : g = 0 that has [ζ · γ ] = nm + s. Since subtracting from g a multiple of f does not change [ζ · γ ], we assume We will need: If in < m − n, by the above, and so and so, this time, leading to the same contradiction as above. Thus, c 0,0 = 0. Now, since c 0,0 = 0, one has o t C ≥ n and so If there is i, in < m, for which d i,0 = 0, then, assuming again that it is the minimal one, still o t D = in and we have once again against the hypothesis. Thus d i,0 = 0 for all i < m/n.
From what we have proved till now, C f x + D f y = −mc 1,0 x m + nd 0,1 y n + . . . , the dots meaning terms of higher twisted degree. If mc 1,0 = nd 0,1 , then direct substitution yields [ζ · γ ] = nm, and so again a contradiction.
Assume to have thus mc 1,0 = nd 0,1 = 0. Then, up to dividing g = C f x + D f y by mc 1,0 , we may assume that c 1,0 = 1/m and d 0,1 = 1/n, after which where the dots indicate terms of twisted degree higher than nω 1 + mω 2 = nm + σ . As it is clear, g has twisted initial form y n − x m and therefore ζ is an irreducible germ with single characteristic exponent m/n that goes through the first point on γ after the last satellite.
If ζ does not go through the point q σ , on γ and in the σ -th neighbourhood of the last satellite, then, by the Noether formula, [ζ · γ ] < nm + σ against the hypothesis. Otherwise, by [7], 4.5, there is an invertible series, necessarily of the form 1 + u, u(0, 0) = 0, such that (1 + u)g and f have the same partial sum of twisted degree nm + σ − 1. Note that the monomials each monomial αx i y j of g gives rise to by multiplication by 1 + u are αx i y j itself, plus monomials α x k y r with (k, r ) = (i, j) + v, v a positive vector. Then, all monomials given rise to by monomials of correspond to non-standing points. Since (ω 1 , ω 2 ) is a standing point, (1 + u)g has the same monomial of bidegree (ω 1 , ω 2 ) as g, and therefore the form the dots still indicating terms of twisted degree higher than nω 1 + mω 2 = nm + σ . Then, and considering the germ ζ : the last contradiction needed in order to prove the claim. Lemma 7.4. Still assume that ζ : g = C f x + D f y = 0 has [ζ · γ ] = nm + s, s > σ . Consider the vector field (C∂ x + D∂ y ), defined in a neighbourhood of the origin, and its associated local flow ϕ u : x * = x * (x, y, u), y * = y * (x, y, u).
Then each local automorphism ϕ u is principal.
Proof of 7.4:. Write By the hypothesis, associated to C∂ x + D∂ y . Assume that is a Puiseux parameterization of γ and substitute it in the series ϕ * where the power series θ(u) is non-zero and the dots indicate terms of higher degree in t. Then r = [ϕ * u (γ ) · γ ] for u = 0 and close enough to 0. Since, by 7.4, ϕ u is principal, it leaves invariant all points on γ up to the first free point after the satellite points and therefore r > nm. The difference r = r − nm is thus a positive integer. For all u close enough to 0, it holds [ϕ * u (γ ) · γ ] ≥ nm + r and therefore ϕ u leaves invariant all points on γ up to the r -th free point p r after the satellite points. We need a further lemma: Proof of 7.5:. By 6.3, for u small enough, θ(u) is the modulus of the translation induced by ϕ * u in the first neighbourhood of p r (relative to an affine coordinate independent of u). After this, by 6.4, it holds for u 1 and u 2 small enough. Then, using just the definition of derivative, θ(u) = [dθ/du] |u=0 u, hence the claim.
End of the proof of 7.1. After taking derivatives with respect to u at u = 0 in (6), This yields nm + r = [ζ · γ ] = nm + s and thus r = s. After this, 6.3 and the equality (6) show that, for u small enough, u = 0, ϕ * u induces a non-identical translation in the first neighbourhood of q s , and therefore ( [7], 5.5) that the coefficients of twisted degree nm + s are irrelevant, as claimed.
For the converse, assume that, for a certain integer s > σ, the coefficients of twisted degree nm + s are irrelevant. Using the notations of [7]. Sect. 12, denote by [ϕ] s the twisted s-jet of a principal automorphism ϕ, by B s the group of twisted s-jets of the principal automorphisms and by W s the subgroup of the twisted s-jets leaving fixed q s . Consider the group homomorphism δ s mapping each twisted s-jet to the modulus of the translation induced by it on the first neighbourhood of q s ( [7], 12.2), namely for v, w admissible of twisted degree at most s. In particular, the zeros of s (X v , Y w ) in W s are the twisted s-jets of the principal automorphims leaving fixed q s+1 .
The coefficients of twisted degree s being irrelevant, δ s is not constant (i. e., not zero) on W s ( [7], 12.8). Being a group homomorphism, δ s , seen as a function on the smooth variety W s , has no critical points, in particular its differential at [I d] s is not zero. We may thus choose an analytic family of twisted s-jetsφ u ∈ W s , defined for u small enough and withφ 0 = [I d] s , which is transverse at [I d] s to the variety of zeros of s in W s . The coefficients up to twisted degree s of the local automorphisms being affine coordinates of the corresponding twisted jet in B s , assume that the above family is given by analytic functions for v, w admissible and with twisted degree at most s. By the transversality above, for u = 0 small enough, and For u small enough, let ϕ u be the representative ofφ u with all its coefficients a v , b w equal to zero for td(v) > nm + s, td(w) > nm + s. Assume it to be given by equalities

y).
For u = 0, the germ ϕ * u (γ ) goes through q s and, by (7), misses q s+1 . Therefore where β(u) = 0 and the dots indicate terms of higher degree in t. Now, by 6.3, both β(u) and (a v (u), b w (u)) are moduli of the action of ϕ u on the first neighbourhood of q s relative to affine coordinates independent of u. Therefore, there is c ∈ C − {0} such that β(u) = c (a v (u), b w (u)) for any u small enough. The above may thus be equivalently written Taking derivatives with respect to u at u = 0 yields This equality, together with (8), shows that the generalized polar has [ζ · γ ] = nm + s, as wanted. If 20 − 63A 2 4,3 + 56A 3,4 = 0, using the Puiseux parameterization of γ it easily turns out that [ζ 0 · γ ] = 52 and therefore, by 7.1, the coefficient A 4,4 is irrelevant, as already seen in [7], 12.14.
The proof of 7.1 provides local automorphisms having a non-trivial action on the first neighbourhood of q s : Given a point p = (i, j) on the Newton plane N , the set will be called the quadrant with vertex p. Remark 7.8. Assume that for a certain p = (i, j) ∈ N there is ζ : g = 0, g ∈ J(γ ) with [ζ · γ ] = td( p). Then, for any p = (i + k, j + r ) ∈ Q p , x k y r g ∈ J(γ ) defines a germ ζ with [ζ ·γ ] = td( p ). In other words, if td( p) is not a gap of J(γ ), neither is td( p ) for any p ∈ Q p . Using 7.1, this directly gives new information regarding relevance of coefficients: Corollary 7.9. Let γ be as above, defined by the equation f = y n − x m + ni+m j>nm A i, j x i y j = 0. If the coefficient A i, j of γ , corresponding to p = (i, j), ni +m j > nm +σ , is irrelevant, then so are all coefficients corresponding to points in Q p .
Still assume that γ has finite Zariski invariant σ and Zariski point ω = (ω 1 , ω 2 ). As in [7], Section 12, take κ = min{td(m − 1, ω 2 ), td(ω 1 , n − 1)} − nm and, in the Newton plane N , let the triangle T be either Then: Remark 7.11. Of course, from the quadrants Q p k of 7.10, those with p k in another quadrant are redundant. In fact, one may choose p 4 to be the lowest twisted degree point in T corresponding to an irrelevant coefficient and, inductively, p k to be the lowest twisted degree point in T − k <k Q p k corresponding to an irrelevant coefficient. Then still U = k=1 Q p k and the decomposition is irredundant.
The border of U in 7.10 is composed of two half-lines on the coordinate axes, with ends (0, n−1) and (m−1, 0), and a stairs-shaped line joining their ends: we will call this line the staircase line of γ , denoted SL(γ ). An irredundant decomposition U = k=1 Q p k being obviously unique, we will call the points p k , k = 1, . . . , , the corners of SL(γ ).
Remark 7.12. After 7.10, the coefficients corresponding to either the Zariski point ω or a point strictly below SL(γ ) are relevant; those corresponding to a point other than ω and placed on or above SL(γ ) are irrelevant.
Remark 7.14. Corollary 7.9 and [7], 12.17 combine to give non-obvious information. Still in the case of [7], 12.18, if A 8,3 is irrelevant, then so is A 9,3 and, as a consequence, A 7,4 is invariant. Similarly, if A 6,4 is irrelevant, so is A 7,4 and then A 9,3 is invariant.
The staircase line of a quasihomogeneous germ may be taken to have corners (m − 1, 0) and (0, n − 1); in such a way, by [7], 7.13, after dropping the mentions to the Zariski point, Remark 7.12 still applies.

Complements regarding the gaps of J(γ )
As seen in 7.1, the gaps d of J(γ ) with d > nm + σ are the twisted degrees of the points corresponding to relevant coefficients. In this section we will determine the gaps d of J(γ ) with d ≤ nm + σ . This will provide a useful characterization of the Zariski invariant σ (γ ).
In this section and the next one we will make frequent use of the fact that, if finite, σ = σ (γ ) satisfies σ + mn < 2nm − n − m + 1, just because the Zariski point is a standing point. Proof. As in former occasions, up to replacing γ with an analytically equivalent germ, we assume all non-standing coefficients of the equation of γ up to the twisted degree nm + σ to be zero, and therefore an equation of γ to be f = y n − x m + A ω 1 ,ω 2 x ω 1 y ω 2 + · · · = 0, A ω 1 ,ω 2 = 0, where (ω 1 , ω 2 ) is the Zariski point of γ and the dots stand for terms of higher twisted degree.
By substituting a Puiseux parameterization of γ it is direct to check that the generalized polar Since all their monomials correspond to non-standing points, neither of them has twisted degree d. Therefore, either two of them or all three cancel.
In the first case, assume for instancē Thenā is a multiple of x m−1 and so after which d = to(c f y ) would be the twisted order of a non-standing point, against the hypothesis. The other two possibilities in this case may be dealt with similarly.
If both sides are not zero thenā y + nc is a non-zero multiple of x m−1 , after which, a computation as above gives again d > nm + σ . Otherwise, In particular, td(b) ≥ n and td(c) ≥ m, after which it is clear that As said, quasihomogeneous germs have little interest, which is the reason why they have been seldom considered before. Anyway, for the sake of completeness, any quasihomogeneous germ τ being analytically equivalent to y n − x m = 0, it is direct to check using the latter that Proof. If the twisted initial formḡ of g has two or more monomials, then each of these monomials corresponds to a non-standing point (by 2.1) and therefore is a multiple of either x m−1 or y n−1 . Otherwiseḡ is a monomial, say corresponding to a point (i, j). Then, by 7.1 and 8.1, either (i, j) is one of the points (m −1, 0), (0, n − 1), ω or ni +m j > nm +σ and the coefficient A i, j of the equation of γ is irrelevant. In any case, by 7.10, (i, j) ∈ Q k for some k andḡ is then a multiple of X k .

Computations and examples
In this section we will present some computation procedures that allow to compute the Zariski invariant σ (γ ) and to examine, for fixed n, m and σ (γ ), the different possibilities of relevance of coefficients through Theorem 7.1. They will be used in three examples. Our procedures are simplified adaptations of procedures that are usual in computational algebra (see for instance [9]); nevertheless, since they are not direct applications, we will provide the -rather simple-arguments needed to support them.
As before, γ is the germ of curve defined by f = y n −x m + ni+m j>nm A i, j x i y j = 0. Assume to have fixed g 1 , . . . , g r ∈ C{x, y}, r ≥ 2, each with a monomial as twisted initial form. Assume furthermore that the twisted initial forms of g 1 and g 2 are scalar multiples of powers of x and y, respectively. For i = 1, . . . , r , let p i be the point corresponding to the twisted initial form of g i , and Q the union of quadrants Q = Q p 1 ∪ · · · ∪ Q p r . To avoid trivialities, we assume also Q p i ⊂ Q p j for i = j. Note that there are finitely many points in N 2 − Q.
Given any g ∈ C{x, y}, substract from g multiples of the g i whose initial forms cancel the monomials of g corresponding to the points in Q of minimal twisted degree; then do the same with the difference and so on, until having cancelled all monomials of g corresponding to points in Q of twisted degree, say, ρ or less. This will give an expression where all the monomials of R correspond to points not in Q, to(K ) > ρ and, clearly, to(h k g k ) ≥ to(g), k = 1, . . . , r , provided h k g k = 0. As soon as ρ is higher than the maximum of the twisted degrees of the points in N 2 − Q, all monomials of the multiples of the g k used for the cancellations correspond to points in Q and therefore R becomes independent of ρ. This satisfied, we will say that the procedure leading to the equality (9) is a division of g by g 1 , . . . , g r ; the h k will be called quotients of the division and R a remainder of dividing g by g 1 , . . . , g r . Neither the quotients nor the remainder are uniquely determined by g and the g k , as it is easy to check. The usual computational algebra division in the local case is different and quite more complicated, see for instance [8], 4.3.
We will make use of the following fact: f y ), and so also to J(γ ).
Proof. Perform a division of g by f x , f y as (9). Note that the points of Q with integral coordinates are the non-standing points. The ideal ( f x , f y ) being (x, y)primary, we take the positive integer ρ such that any series with twisted order higher than ρ belongs to ( f x , f y ), and so, in particular, so does the complementary term K . Since g has twisted order 2nm − n − m + 1 or higher, the same holds for any of the multiples of f x or f y used to cancel monomials in the division procedure.
On the other hand, any standing monomial has twisted degree strictly less than 2nm − n − m + 1 and therefore no standing monomial does appear in g or along the division procedure, which forces the remainder R to be zero. Then the claim follows from equality (9).
Remark 9.2. In our applications we will always take g 1 = f x and g 2 = f y , in which case all the non-standing points belong to Q. Also, unless otherwise said, we will take ρ = 2nm − n − m + 1: then all points p with td( p) ≥ ρ are non-standing, and therefore belong to Q; furthermore, K ∈ J(γ ) by 9.1.
The next proposition provides an easy way of computing the Zariski invariant σ (γ ). Of course, it is useful only in case the equation has some non-zero nonstanding monomial, as otherwise the Zariski invariant may be directly read from the equation, see Section 3. Proof. The points of Q with integral coordinates are the non-standing points, and so all the monomials of R are standing monomials. In particular, if R = 0, to(R) is the twisted degree of a standing point. By the choices of 9.2 R ∈ J(γ ). Then, if γ is quasihomogeneous, using 5.3 contradicts 8. 3.
Assume now that γ is not quasihomogeneous. Then, by 5.3 and 8.2, there is g ∈ J(γ ) with to(g) = nm + σ . Assume Using the division of f by f x , f y gives an equality where b , c , U ∈ C{x, y} and to(U ) ≥ 2nm − n − m + 1 > to(g). If the twisted initial forms of b f x and c f y , namely −mb x m−1 and nc y n−1 , cancel, then, arguing as in the proof of 8.1 to and so to(g) = to(a) + to(R).
Otherwise, the twisted initial form of b f x + c f y is a sum of non-standing monomials. Therefore to(b f x + c f y ) = to(g) and, again, to(g) = to(a) + to(R).
Thus, in both cases, by the minimality in 8.2, to(a) = 0 and the claim follows. Example 9.4. As in [7], 7.12, take γ defined by f = y 5 − x 7 − 7x 6 y − 21x 5 y 2 . A division of f by its derivatives is Hence, the Zariski invariant is 6 and so the Zariski point is (4,3). Note the monomial −21x 5 y 2 , which is standing, appears in the equation of γ , has twisted degree 39 < 41 = td(4, 3) and does not correspond to the Zariski point. On the other hand, no non-zero monomial of f corresponds to the Zariski point.
The Tjurina number of γ is an analytic invariant, usually denoted τ (γ ), which may be defined by the rule τ (γ ) = dim C C{x, y}/J(γ ). As one can expect in view of 8.4, τ (γ ) can be directly read from SL(γ ): Proof. Write L = N 2 − k=1 Q p k and, for any p = (i, j) ∈ N 2 , X p = x i y j . We will prove that the classes of the X p , p ∈ L, make a basis of C{x, y}/J(γ ). Indeed, in case of they being linearly dependent, it would be g = p∈L a p X p ∈ J(γ ) for some a p ∈ C, not all zero; then, g = 0 andḡ = a p X p for some p ∈ L (due to 2.1), which contradicts 8.4. To prove that the classes of the X p , p ∈ L, generate the quotient, take g k ∈ J(γ ) withḡ k a monomial corresponding to the corner p k , k = 1, . . . , . Then any g ∈ C{x, y} is congruent mod. J(γ ) to a remainder of its division by p 1 , . . . , p .
The above, together with Remark 7.13, may explain the difficulties in computing the minimal Tjurina number for a given equisingularity class: recursive procedures were given in [3] and [15], and only recently a closed formula has been given in [1].
Lemma 9.7. If g 1 , g 2 ∈ C{x, y} − {0} and g k = M k g k , M k a non-zero monomial, k = 1, 2, then both g 1 , g 2 have monomials as twisted initial forms and where M is a non-zero monomial.
Proof. Direct from the definition of [ * ].
Lemma 9.8. Assume that the twisted initial forms of g k ∈ C{x, y} − {0}, k = 1, . . . , r, r ≥ 2, are all monomials of the same twisted degree ρ. If to( r k=1 g k ) > ρ, then there exist a i, j ∈ C for which Proof. The hypothesis is that the initial forms of the g k cancel: r k=1ḡ k = 0. Then the same happens if the summation is restricted to the g k whose initial forms are scalar multiples of an arbitrarily fixed monomial M. Therefore, by splitting the given set {g 1 , . . . , g r }, it suffices to prove the claim with the supplementary hypothesis that all theḡ k are scalar multiples of the same monomial, sayḡ k = b k M, k = 1, . . . , r . Then r k=1 b k = 0 and [g i * g j ] = b j g i − b i g j , after which it is enough to check that it holds Proof. Obviously, the initial forms of g 1 = f x and g 2 = f y are monomials. By 9.3, g 3 = 0 and its twisted initial form is a monomial corresponding to the Zariski point. Assume inductively that for a given k, 3 ≤ k < , g 1 , . . . , g k are non-zero and have monomials as twisted initial forms. Then g k+1 is defined and non-zero because k < . Since Q p 1 ∪ Q p 2 is the set of the non-standing points, the union of quadrants U k = Q p 1 ∪ · · · ∪ Q p k , used in the division whose remainder is g k+1 , contains all the non-standing points. Then all non-zero monomials of the remainder g k+1 are standing monomials and no two have the same twisted degree, by 2.1. This in particular assures that the twisted initial form of g k+1 is a monomial. As far as option (a) does not occur, the sets N 2 − U k are finite and make an strictly decreasing sequence. Having N 2 − U k = ∅ obviously forces R r,s = 0 for 1 ≤ r < s ≤ k, hence the finiteness of the procedure.
Fix k, 3 ≤ k ≤ and assume to have chosen g 1 , . . . , g k as in the claim. Take = min{to(R r,s )} 1≤r <s≤k , ≤ ∞. We will prove that no g ∈ J(γ ) with initial form a monomial corresponding to a point in N 2 − U k has to(g) < . This proved, on one hand, for 3 ≤ k < , p k+1 is the point of minimal twisted degree in N 2 − U k corresponding to the twisted initial form of an element of J(γ ). On the other, no point in N 2 − U corresponds to the twisted initial form of an element of J(γ ).
Assume thus that the initial form of g ∈ J(γ ) is a monomial whose corresponding point is p ∈ N 2 − U k , and also that to(g) < . Note that then p is the only point in N 2 with td( p) = to(g), by 2.1. Since g 1 , . . . , g k generate J(γ ), there is at least an expression Then take δ = min{to(h 1 g 1 ), . . . , to(h g } and note that δ ≤ . Then, among all g as above and all their expressions as (10), choose a pair for which δ is maximal and still use for them the notations as in (10). Up to reordering, assume that to(h i g i ) = δ for i = 1, . . . , s and to(h i g i ) > δ for i = s + 1, . . . , k. Split We will fix our attention on g , just retaining from g that all its summands have twisted order strictly higher than δ; in particular, to(g ) > δ . Just because of this, it cannot be to(g ) = δ, as then to(g) = δ would be the twisted order of a point in U k . Decompose eachh i , 1 ≤ i ≤ s, into the sum of its monomials, Since, as noted above, to(g ) > δ, by 9.8 there are a i, j,i , j ∈ C for which Using 9.7 this yields where each M i, j,i , j is a monomial and td(M i, j,i , j ) = to([h i, j g i * h i , j g i ]) − to([g i * g i ]) > δ − to([g i * g i ]). (13) Write the chosen division of [g i * g i ] by g 1 , . . . , g k in the form where R i,i is the remainder, B i,i a sum of multiples of g 1 , . . . , g k , each with twisted order non-less than the twisted order of [g i * g i ], and K i,i a series with twisted order at least 2nm − n − m + 1. By replacing in (12) and going back to (11) we get Our hypothesis regarding to(g) forces to(g) < to( then, necessarily, to(ĝ) = to(g), which in turn forces g andĝ to have the same monomial as initial form (by 2.1). Obviously,ĝ ∈ J(γ ). In addition, as noticed when defining it, any of the multiples of the g i whose sum is B i,i has twisted order to([g i * g i ]) or higher. Using (13), any of the multiples of the g i whose sum is M i, j,i , j B i,i has twisted order strictly higher than δ. The same being true for g , g contradicts our choice of g and its expression.
The series g k of 9.9 may be computed using any computer algebra system. In practice, many of the [g i * g k ] involved may be discarded a priori due to a too high cancelled twisted degree. Coefficients that are continuous or conditional invariants may be taken as free parameters in order to discuss the different possibilities; in such a case the twisted initial forms of the g i , i > 3, provide the conditions for the relevance of the conditional invariants. Next are two examples. The first one already appeared in [16]  The first equality confirms the Zariski point, already evident from the equation. The second one shows that the coefficients corresponding to (16,6) and (17,6) are continuous invariants (both with value zero) for all values of the invariant A. The coefficient corresponding to (18,6) is irrelevant for A = −19/54, while it is also a continuous invariant for A = −19/54. Figure 4 shows the staircase lines for both cases; they in particular explain the jumping of the Tjurina number, from 153 to 154, for A = −19/54.
The reader may have noticed how the coefficient corresponding to (18,6) plays an important role in example 9.4, even if its value is zero in all cases. This shows the convenience of considering analytic relevance rather than just the possibility of turning a coefficient into zero by an analytic automorphism (elimination of coefficients), which may be confusing.
Then, −162 A 2 9,3 + 1215A 15,1 A 2 9,3 − 729A 4 9,3 − 6561 8 A 6 9,3 )x 14 y 3 + . . . (15) showing that A 13,3 is necessarily invariant. This time the invariance is not due to the constraints of [7] 12.7; actually, A 18,3 cannot be irrelevant because its twisted degree is lesser than the cancelled twisted degrees appearing in the computation of g 5 . The reader may note that, due to the invariance of A 18,3 and taking in account the restriction (14), the number of free continuous invariants is 8 in both Case 2.1 and the "generic" Case 1. Still in Case 2.1 and depending on the coefficient in (15) Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/ licenses/by/4.0/.
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