On the Milnor formula in arbitrary characteristic
aa r X i v : . [ m a t h . AG ] D ec On the Milnor formula in arbitrary characteristic
Evelia R. Garc´ıa Barroso and Arkadiusz P loskiDecember 18, 2018
Dedicated to Antonio Campillo on the occasion of his 65th birthday
Abstract
The Milnor formula µ = 2 δ − r + 1 relates the Milnor number µ , thedouble point number δ and the number r of branches of a plane curvesingularity. It holds over the fields of characteristic zero. Melle andWall based on a result by Deligne proved the inequality µ ≥ δ − r + 1in arbitrary characteristic and showed that the equality µ = 2 δ − r + 1characterizes the singularities with no wild vanishing cycles. In thisnote we give an account of results on the Milnor formula in charac-teristic p . It holds if the plane singularity is Newton non-degenerate(Boubakri et al. Rev. Mat. Complut. (2010) 25) or if p is greaterthan the intersection number of the singularity with its generic po-lar (Nguyen H.D., Annales de l’Institut Fourier, Tome 66 (5) (2016)).Then we improve our result on the Milnor number of irreducible singu-larities (Bull. London Math. Soc. 48 (2016)). Our considerations arebased on the properties of polars of plane singularities in characteristic p . Introduction
John Milnor proved in his celebrated book [17] the formula µ = 2 δ − r + 1 , (M) Mathematics Subject Classification:
Primary 14H20; Sec-ondary 32S05.Key words and phrases: plane singularity, Milnor number, degreeof conductor, factorization of polar curve.The first-named author was partially supported by the SpanishProject MTM 2016-80659-P. µ is the Milnor number µ , δ the double point number and r the numberof branches of a plane curve singularity. The Milnor’s proof of (M) is basedon topological considerations. A proof given by Risler [21] is algebraic andshows that (M) holds in characteristic zero.On the other hand Melle and Wall based on a resultd by Deligne [5] provedthe inequality µ ≥ δ − r + 1 in arbitrary characteristic and showed that theMilnor formula holds if and only if the singularity has not wild vanishingcycles [16]. In the sequel we will call a tame singularity any plane curvesingularity verifying (M).Recently some papers on the singularities satisfying (M) in characteristic p appeared. In [1]) the authors showed that planar Newton non-degeneratesingularities are tame. Different notions of non-degeneracy for plane curvesingularities are discussed in [10]. In [18] the author proved that if thecharacteristic p is greater than the kappa invariant then the singularity istame. In [7] and [11] the case of irreducible singularities is investigated. Ouraim is to give an account of the above-mentioned results.In Section 1 we prove that any semi-quasihomogeneous singularity is tame.Our proof is different from that given in [1] and can be extended to the case ofKouchnirenko nondegenerate singularities ([1, Theorem 9]). In Section 2 and3 we generalize Teissier’s lemma ([22, Chap. II, Proposition 1.2]) relating theintersection number of the singularity with its polar and the Minor numberto the case of arbitrary characteristic and reprove the result due to H.D.Nguyen [18, Corollary 3.2] in the following form: if p > µ ( f ) + ord( f ) − Let K be an algebraically closed field of characteristic p ≥
0. For anyformal power series f ∈ K [[ x, y ]] we denote by ord( f ) (resp. in( f )) the order (resp. the initial form of f ). A power series l ∈ K [[ x, y ]] is calleda regular parameter if ord( l ) = 1. A plane curve singularity (in short: asingularity ) is a nonzero power series f of order greater than one. For anypower series f, g ∈ K [[ x, y ]] we put i ( f, g ) := dim K K [[ x, y ]] / ( f, g ) andcalled it the intersection number of f and g . The Milnor number of f is µ ( f ) := dim K K [[ x, y ]] / (cid:18) ∂f∂x , ∂f∂y (cid:19) .
2f Φ is an automorphism of K [[ x, y ]] then µ ( f ) = µ (Φ( f )) (see [1, p. 62]). Ifthe characteristic of K is p = char K > µ ( f ) = + ∞ and µ ( uf ) < + ∞ for a unit u ∈ K [[ x, y ]] (take f = x p + y p − and u = 1 + x ).Let f ∈ K [[ x, y ]] be a reduced (without multiple factors) power series andconsider a regular parameter l ∈ K [[ x, y ]]. Assume that l does not divide f .We call the polar of f with respect to l the power series P l ( f ) = ∂ ( f, l ) ∂ ( x, y ) = ∂f∂x ∂l∂y − ∂f∂y ∂l∂x . If l = − bx + ay for ( a, b ) = (0 ,
0) then P l ( f ) = a ∂f∂x + b ∂f∂y .For any reduced power series f we put O f = K [[ x, y ]] / ( f ), O f the integralclosure of O f in the total quotient ring of O f and δ ( f ) = dim K O f / O f (thedouble point number). Let C be the conductor of O f , that is the largestideal in O f which remains an ideal in O f . We define c ( f ) = dim K O f / C (the degree of conductor ) and r ( f ) the number of irreducible factors of f .The semigroup Γ( f ) associated with the irreducible power series f is definedas the set of intersection numbers i ( f, h ), where h runs over power seriessuch that h f ).The degree of conductor c ( f ) is equal to the smallest element c of Γ( f ) suchthat c + N ∈ Γ( f ) for all integers N ≥ f we define µ ( f ) := c ( f ) − r ( f ) + 1 . In particular, if f is irreducible then µ ( f ) = c ( f ). Proposition 1.1
Let f = f · · · f r ∈ K [[ x, y ]] be a reduced power series,where f i is irreducible for i = 1 , . . . , r . Then(i) µ ( f ) = µ ( uf ) for any unit u of K [[ x, y ]] .(ii) µ ( f ) + r − r X i =1 µ ( f i ) + 2 X ≤ i Property ( i ) is obvious. To check ( ii ) observe that r X i =1 µ ( f i )+2 X ≤ i 1. Then by ( ii ) we get µ ( f ) + r − ≥ X ≤ i 0, which proves ( v ).4 emark 1.2 Using Proposition 1.1 (ii) we check the following property:Let f = g · · · g s ∈ K [[ x, y ]] be a reduced power series, where the power series g i for i = 1 , . . . , s are pairwise coprime. Then µ ( f ) + s − s X i =1 µ ( g i ) + 2 X ≤ i 0) and (0 , n ). Let d = gcd( m, n ). Thenin −→ w ( f ) = F ( x m/d , y n/d ) , where F ( u, v ) ∈ K [ u, v ] is a homogeneous polyno-mial of degree d . Proposition 1.3 Suppose that in −→ w ( f ) has no multiple factors. Then µ ( f ) = (cid:18) ord −→ w ( f ) n − (cid:19) · (cid:18) ord −→ w ( f ) m − (cid:19) . roof. In the proof we will use lemmas collected in the Appendix.Observe that if in −→ w ( f ) has no multiple factors then in −→ w ( f ) = m −→ w ( f ) (in −→ w ( f )) o ,where m −→ w ( f ) ∈ { , x, y, xy } and (in −→ w ( f )) o is a convenient power series or aconstant. To prove the proposition we will use Hensel’s Lemma (see Lemma5.3) and Remark 1.2. We have to consider several cases.Case 1: in −→ w ( f ) = (const) · x or in −→ w ( f ) = (const) · y .In this case ord( f ) = 1 and by Proposition 1.1 (v) µ ( f ) = 0. If in −→ w ( f ) =(const) · x (resp. in −→ w ( f ) = (const) · y ) then ord −→ w ( f ) = n (resp. ord −→ w ( f ) = m ) and (cid:18) ord −→ w ( f ) n − (cid:19) (cid:18) ord −→ w ( f ) m − (cid:19) = 0 . Case 2: in −→ w ( f ) = (const) · xy. By Hensel’s Lemma (see Lemma 5.3) f = f f , where in −→ w ( f ) = c x ,in −→ w ( f ) = c y with constants c , c = 0. Using Remark 1.2 and Lemma5.1 we get µ ( f ) + 1 = µ ( f f ) + 1 = µ ( f ) + µ ( f ) + 2 i ( f , f ) = 0 + 0 + 2 . µ ( f ) = 1. On the other hand ord −→ w ( f ) = n + m and (cid:18) ord −→ w ( f ) n − (cid:19) (cid:18) ord −→ w ( f ) m − (cid:19) = 1 . Case 3: The power series in −→ w f is convenient.Assume additionally that the line nα + mβ = ord −→ w ( f ) intersects the axesin points ( m, 0) and (0 , n ). Let d = gcd( n, m ). Then the −→ w -initial form of f is in −→ w f = d Y i =1 (cid:16) a i x m/d + b i y n/d (cid:17) , where a i x m/d + b i y n/d are pairwise coprime. By Hensel’s Lemma (see Lemma5.3) we get a factorization f = Q di =1 f i , where in −→ w f i = a i x m/d + b i y n/d for i = 1 , . . . , d . The factors f i are irreducible with semigroup Γ( f i ) = md N + nd N and µ ( f i ) = c ( f i ) = (cid:16) md − (cid:17) (cid:16) nd − (cid:17) i ( f i , f j ) = ord −→ w f i ord −→ w f j mn = mnd , for i = j and we get by Proposition 1.1 (ii) µ ( f )+ d − d X i =1 µ ( f i )+2 X ≤ i 1) = (cid:16) ord −→ w fn − (cid:17) (cid:16) ord −→ w fm − (cid:17) , since theweighted order of f is ord −→ w f = mn .Now consider the general case, that is when the line nα + mβ = ord −→ w ( f ) in-tersects the axes in points ( m , 0) = (cid:16) ord −→ w fn , (cid:17) and (0 , n ) = (cid:16) , ord −→ w ( f ) m (cid:17) .Then f is semi-quasihomogeneous with respect to −→ w = ( n , m ) and theline n α + m β = ord −→ w ( f ) intersects the axes in points ( m , 0) and (0 , n ).By the first part of the proof we get µ ( f ) = ( m − n − 1) = (cid:18) ord −→ w ( f ) n − (cid:19) (cid:18) ord −→ w ( f ) m − (cid:19) , which proves the proposition in Case 3.Case 4: in −→ w ( f ) = x (in −→ w ( f )) o or in −→ w ( f ) = y (in −→ w ( f )) o , where (in −→ w ( f )) o isconvenient.This case follows from Hensel’s Lemma (Lemma 5.3), Case 1 and Case 3.Case 5: in −→ w ( f ) = xy (in −→ w ( f )) o , where (in −→ w ( f )) o is convenient.This case follows from Hensel’s Lemma (Lemma 5.3), Case 2 and Case 3. Theorem 1.4 Suppose that in −→ w ( f ) has no multiple factors. Then f is tameif and only if f is a semi-quasihomogeneous singularity with respect to −→ w . Proof. We have µ ( f ) = (cid:16) ord −→ w ( f ) n − (cid:17) (cid:16) ord −→ w ( f ) m − (cid:17) by Proposition 1.3.On the other hand, by Lemma 5.2, we get that µ ( f ) = (cid:16) ord −→ w ( f ) n − (cid:17) (cid:16) ord −→ w ( f ) m − (cid:17) 7f and only if the system of equations ∂∂x in −→ w ( f ) = 0 , ∂∂y in −→ w ( f ) = 0has the only solution ( x, y ) = (0 , (iv) . Example 1.5 Let f ( x, y ) = x m + y n + P αn + βm>nm c α β x α y β and let d =gcd( m, n ) . Then in −→ w ( f ) = x m + y n has no multiple factors if and only if d (mod p ). If d (mod p ) then f is tame if and only if m (mod p ) and n (mod p ). Corollary 1.6 The semi-quasihomogeneous singularities are tame. Corollary 1.6 is a particular case of the following Theorem 1.7 (Boubakri, Greuel, Markwig [1, Theorem 9]). The planarNewton non-degenerate singularities are tame. p ≥ The intersection theoretical approach to the Milnor number in characteristiczero [4] is based on a lemma due to Teissier who proved a more general result(the case of hypersurfaces) in [22, Chapter II, Proposition 1.2]. A generalformula on isolated complete intersection singularity is due to Greuel [8] andLˆe [14]. In this section we study Teissier’s Lemma in arbitrary characteristic p ≥ f ∈ K [[ x, y ]] be a reduced power series and l ∈ K [[ x, y ]] be a regularparameter. Assume that l does not divide f and consider the polar P l ( f ) = ∂f∂x ∂l∂y − ∂f∂y ∂l∂x of f with respect to l . In this section we assume, without lossof generality, that ord( l (0 , y )) = 1. Lemma 2.1 Let f ∈ K [[ x, y ]] be a reduced power series and l ∈ K [[ x, y ]] be a regular parameter. Then i ( l, P l ( f )) ≥ i ( f, l ) − with equality if andonly if i ( f, l ) p ) . Proof. Recall that ord( l (0 , y )) = 1. Let φ ( t ) = ( φ ( t ) , φ ( t )) be a goodparametrization of the curve l ( x, y ) = 0 (see [19, Section 2]. In particular8 = l ( φ ( t )) so ddt l ( φ ( t )) = 0. On the other hand we have ord( φ ( t )) = i ( x, l ) = ord( l (0 , y )) = 1 and φ ′ (0) = 0. Differentiating f ( φ ( t )) and l ( φ ( t ))we get ddt f ( φ ( t )) = ∂f∂x ( φ ( t )) φ ′ ( t ) + ∂f∂y ( φ ( t )) φ ′ ( t ) (1)and 0 = ddt l ( φ ( t )) = ∂l∂x ( φ ( t )) φ ′ ( t ) + ∂l∂y ( φ ( t )) φ ′ ( t ) . (2)From (2) we have ∂l∂x ( φ ( t )) φ ′ ( t ) = − ∂l∂y ( φ ( t )) φ ′ ( t ) and by (1) and the defi-nition of P l ( f ) we get P l ( f )( φ ( t )) φ ′ ( t ) = ddt f ( φ ( t )) ∂l∂y ( φ ( t )) . Since φ ′ ( t ) and ∂l∂y ( φ ( t )) are units in K [[ t ]] we haveord( P l ( f )( φ ( t ))) = ord (cid:18) ddt f ( φ ( t )) (cid:19) ≥ ord( f ( φ ( t ))) − , with equality if and only if ord( f ( φ ( t ))) p ). Now the lemma followsfrom the formula i ( h, l ) = ord( h ( φ ( t ))) which holds for every power series h ∈ K [[ x, y ]]. Corollary 2.2 Suppose that i ( f, l ) = ord( f ) p ) for a regularparameter l ∈ K [[ x, y ]] . Then(a) i ( l, P l ( f )) = ord( f ) − ,(b) ord( P l ( f )) = ord( f ) − ,(c) if h is an irreducible factor of P l ( f ) then i ( l, h ) = ord( h ) . Proof. Property ( a ) follows immediately from Lemma 2.1. To check ( b )observe that we get ord( P l ( f )) = ord( P l ( f )) · ord( l ) ≤ i ( l, P l ( f )) = ord( f ) − 1, where the last equality follows from ( a ). The inequality ord( P l ( f )) ≥ ord( f ) − P l ( f ) = Q si =1 h i , where h i is irreducible. From ( a ) and ( b ) we get0 = i ( l, P l ( f )) − ord( P l ( f )) = s X i =1 ( i ( l, h i ) − ord( h i )) . Since i ( l, h i ) ≥ ord( h i ) we have i ( l, h i ) = ord( h i ) for i = 1 , . . . , s whichproves ( c ). 9 roposition 2.3 (Teissier’s Lemma in characteristic p ). Let f ∈ K [[ x, y ]] be a reduced power series. Suppose that(i) i ( f, l ) p ) ,(ii) for any irreducible factor h of P l ( f ) we get i ( l, h ) p ) .Then i ( f, P l ( f )) ≤ µ ( f ) + i ( f, l ) − with equality if and only if(iii) for any irreducible factor h of P l ( f ) we get i ( f, h ) p ) . Proof. Fix an irreducible factor h of P l ( f ) and let ψ ( t ) = ( ψ ( t ) , ψ ( t ))be a good parametrization of the curve h ( x, y ) = 0. Then ord( l ( ψ ( t ))) = i ( l, h ) p ) by ( ii ) and ord (cid:0) ddt l ( ψ ( t )) (cid:1) = ord( l ( ψ ( t ))) − 1. Differen-tiating f ( ψ ( t )) and l ( ψ ( t )) we get ddt f ( ψ ( t )) = ∂f∂x ( ψ ( t )) ψ ′ ( t ) + ∂f∂y ( ψ ( t )) ψ ′ ( t ) , (3)and ddt l ( ψ ( t )) = ∂l∂x ( ψ ( t )) ψ ′ ( t ) + ∂l∂y ( ψ ( t )) ψ ′ ( t ) . (4)Since P l ( f )( ψ ( t )) = 0, it follows from (3) and (4) that ddt f ( ψ ( t )) ∂l∂y ( ψ ( t )) = ddt l ( ψ ( t )) ∂f∂y ( ψ ( t )) . (5)Since ∂l∂y ( ψ ( t )) is a unit in K [[ t ]], taking orders in (5) we haveord( f ( ψ ( t ))) − ≤ ord (cid:18) ddt f ( ψ ( t )) (cid:19) = ord (cid:18) ddt l ( ψ ( t )) (cid:19) + ord (cid:18) ∂f∂y ( ψ ( t )) (cid:19) = ord( l ( ψ ( t ))) − (cid:18) ∂f∂y ( ψ ( t )) (cid:19) , where the last equality follows from ord( l ( ψ ( t ))) p ).Hence i ( f, h ) ≤ i ( l, h ) + i (cid:16) ∂f∂y , h (cid:17) . Summing up over all h counted with multiplicities as factors of P l ( f ) weobtain 10 ( f, P l ( f )) ≤ i ( l, P l ( f )) + i (cid:18) ∂f∂y , P l ( f ) (cid:19) . (6)By Lemma 2.1 and assumption ( i ) we have i ( l, P l ( f )) = i ( f, l ) − . More-over i (cid:16) ∂f∂y , P l ( f ) (cid:17) = µ ( f ) since ord( l (0 , y )) = 1 and we get from the equal-ity (6) i ( f, P l ( f )) ≤ µ ( f ) + i ( f, l ) − . The equality holds if and only if i ( f, h ) = i ( l, h ) + i (cid:16) ∂f∂y , h (cid:17) for every h ,which is equivalent to the condition i ( f, h ) p ), since i ( f, h ) p ) if and only if ord (cid:0) ddt f ( ψ ( t )) (cid:1) = ord( f ( ψ ( t ))) − Corollary 2.4 (Teissier [22, Chapter II, Proposition 1.2]). If char K = 0 then i ( f, P l ( f )) = µ ( f ) + i ( f, l ) − . Corollary 2.5 Suppose that p = char K > ord( f ) and let i ( f, l ) = ord( f ) .Then i ( P l ( f ) , f ) ≤ µ ( f ) + i ( f, l ) − . The equality holds if and only if for any irreducible factor h of P l ( f ) we get i ( f, h ) p ) . Proof. If ord( f ) < p then i ( f, l ) = ord( f ) p ) and by Corollary2.2 for any irreducible factor h of P l ( f ) we get i ( l, h ) = ord( h ) ≤ ord( P l ( f )) = ord( f ) − < p. Hence i ( l, h ) p ) and the corollary follows from Proposition 2.3. Example 2.6 Let f = x p +2 + y p +1 + x p +1 y , where p = char K > . Take l = y . Then i ( f, l ) = p + 2 p ) , P l ( f ) = ∂f∂x = x p (2 x + y ) and the irreducible factors of P l ( f ) are h = x and h = 2 x + y . Clearly i ( l, h ) = i ( l, h ) = 1 p ) . Moreover i ( f, h ) = i ( f, h ) = p + 1 and all assumptions of Proposition 2.3 are satisfied.Hence i ( f, P l ( f )) = µ ( f )+ i ( f, l ) − and µ ( f ) = i ( f, P l ( f )) − i ( f, l )+1 = p ( p + 1) . Note that l = 0 is a curve of maximal contact with f = 0 . Let l = x . Then i ( f, l ) = ord( f ) = p + 1 , P l ( f ) = − ( y p + x p +1 ) and h = y p + x p +1 is the only irreducible factor of the polar P l ( f ) . Since i ( l , h ) = p , the condition ( ii ) of Proposition 2.3 is not satisfied. However, i ( f, P l ( f )) = µ ( f ) + i ( f, l ) − , which we check directly. Tame singularities Assume that f is a plane curve singularity. Proposition 3.1 Let f = f · · · f r ∈ K [[ x, y ]] be a reduced power series,where f i is irreducible for i = 1 , . . . , r . Suppose that there exists a regularparameter l such that i ( f i , l ) p ) for i = 1 , . . . , r . Then f is tame ifand only if Teissier’s lemma holds, that is if i ( f, P l ( f )) = µ ( f )+ i ( f, l ) − . Proof. By Proposition 1.1 ( iii ) we have that i ( f, P l ( f )) = µ ( f )+ i ( f, l ) − i ( f, P l ( f )) = µ ( f ) + i ( f, l ) − µ ( f ) = µ ( f ). We finishthe proof using Proposition 1.1 ( iv ). Proposition 3.2 (Milnor [17], Risler [21]). If char K = 0 then any planesingularity is tame. Proof. Teissier’s Lemma holds by Corollary 2.4 . Use Proposition 3.1. Proposition 3.3 Let p = char K > . Suppose that p > ord( f ) . Let l be aregular parameter such that i ( f, l ) = ord( f ) . Then f is tame if and only iffor any irreducible factor h of P l ( f ) we get i ( f, h ) p ) . Proof. Take a regular parameter l such that i ( f, l ) = ord( f ). By hy-pothesis we get i ( f, l ) < p so i ( f, l ) p ). By Corollary 2.2 theassumption ( ii ) of Proposition 2.3 is satisfied.Hence i ( f, P l ( f )) ≤ µ ( f )+ i ( f, l ) − i ( f, h ) p ) for any irreducible factor h of P l ( f ). Use Proposition 3.1. Proposition 3.4 (Nguyen [18]). Let p = char K > . Suppose that thereexists a regular parameter l such that i ( f, l ) = ord( f ) and i ( f, P l ( f )) < p .Then f is tame. Proof. We have p > i ( f, P l ( f )) ≥ ord( f ) · ord( P l ( f )). Hence p > ord( f )and we may apply Proposition 3.3. Since i ( f, P l ( f )) < p for any irreduciblefactor h of P l ( f ) we have that i ( f, h ) < p and obviously i ( f, h ) p ). The proposition follows from Proposition 3.3. Theorem 3.5 (Nguyen [18]). If p > µ ( f ) + ord( f ) − then f is tame. Proof. Since f is a singularity we get µ ( f ) > p > µ ( f ) − f ) ≥ ord( f ). By thefirst part of the proof of Proposition 3.3 we have i ( f, P l ( f )) ≤ µ ( f ) +ord( f ) − 1, where l is a regular parameter such that i ( f, l ) = ord( f ). Hence i ( f, P l ( f )) < p and the theorem follows from Proposition 3.4.12 The Milnor number of plane irreducible singu-larities Let f ∈ K [[ x, y ]] be an irreducible power series of order n = ord( f ) and letΓ( f ) be the semigroup associated with f = 0.Let β , . . . , β g be the minimal sequence of generators of Γ( f ) defined by theconditions • β = min(Γ( f ) \{ } ) = ord( f ) = n , • β k = min(Γ( f ) \ N β + · · · + N β k − ) for k ∈ { , . . . , g } , • Γ( f ) = N β + · · · + N β g .Let e k = gcd( β , . . . , β k ) for k ∈ { , . . . , g } . Then n = e > e > · · · e g − >e g = 1. Let n k = e k − /e k for k ∈ { , . . . , g } . We have n k > k ∈{ , . . . , g } and n = n · · · n g . Let n ∗ = max( n , . . . , n g ). Then n ∗ ≤ n withequality if and only if g = 1.The following theorem is a sharpened version of the main result of [7]. Theorem 4.1 Let f ∈ K [[ x, y ]] be an irreducible power series of order n > and let β , . . . , β g be the minimal system of generators of Γ( f ) . Suppose that p = char K > n ∗ . Then the following two conditions are equivalent:(i) β k p ) for k ∈ { , . . . , g } , (ii) f is tame. In [7] the equivalence of ( i ) and ( ii ) is proved under the assumption that p > n .If f ∈ K [[ x, y ]] is an irreducible power series then we get ord( f ( x, f ) or ord( f (0 , y )) = ord( f ). In the sequel we assume that ord( f (0 , y )) =ord( f ) = n . The proof of Theorem 4.1 is based on Merle’s factorizationtheorem: Theorem 4.2 (Merle [15], Garc´ıa Barroso-P loski [7]). Suppose that ord( f (0 , y )) = ord( f ) = n p ) . Then ∂f∂y = h · · · h g in K [[ x, y ]] , where(a) ord( h k ) = ne k − ne k − for k ∈ { , . . . , g } .(b) If h ∈ K [[ x, y ]] is an irreducible factor of h k , k ∈ { , . . . , g } , then b1) i ( f,h ) ord ( h ) = e k − β k n , and(b2) ord( h ) ≡ (cid:16) mod ne k − (cid:17) . Lemma 4.3 Suppose that p > n ∗ . Then i (cid:16) f, ∂f∂y (cid:17) ≤ µ ( f ) + ord( f ) − with equality if and only if β k p ) for k ∈ { , . . . , g } . Proof. Obviously n k p ) for k = 1 , . . . , g and n = n · · · n g p ). Let h be an irreducible factor of ∂f∂y . Then, by Corollary 2.2( c ) i ( h, x ) = ord( h ). By Theorem 4.2 (b2) ord( h ) = m k ne k − , for an index k ∈{ , . . . , g } , where m k ≥ m k ne k − = ord( h ) ≤ ord( h k ) = ne k − ( n k − 1) and m k ≤ n k − < n k < p , which implies m k p ) andord( h ) p ). By Proposition 2.3 we get i (cid:16) f, ∂f∂y (cid:17) ≤ µ ( f )+ord( f ) − (b1) we have the equalities i ( f, h ) = (cid:16) e k − β k n (cid:17) ord( h ) = m k β k and we get i ( f, h ) p ) if and only if β k p ), whichproves the second part of Lemma 4.3. Proof of Theorem 4.1 Use Lemma 4.3 and Proposition 3.1. Example 4.4 Let f ( x, y ) = ( y + x ) + x y . Then f is irreducible and itssemigroup is Γ( f ) = 4 N + 6 N + 13 N . Here e = 4 , e = 2 , e = 1 and n = n = 2 . Hence n ∗ = 2 .Let p > n ∗ = 2 . If p = char K = 3 , then f is tame. On the other handif p = 2 then µ ( f ) = + ∞ since x is a common factor of ∂f∂y and ∂f∂x . Hence f is tame if and only if p = 2 , , . Note that for any l with ord( l ) = 1 wehave i ( f, l ) ≡ . Proposition 4.5 If Γ( f ) = β N + β N then f is tame if and only if β p ) and β p ) . Proof. Let −→ w = ( β , β ). There exists a system of coordinates x, y suchthat we can write f = y β + x β + terms of weight greater than β β . Theproposition follows from Theorem 1.4 (see also [7, Example 2]).In [11] the authors proved, without any restriction on p = char K , thefollowing profound result: Theorem 4.6 (Hefez, Rodrigues, Salom˜ao [11], [12]). Let Γ( f ) = β N + · · · + β g N . If β k p ) for k = 0 , . . . , g then f is tame. The question as to whether the converse of Theorem 4.6 is true remainsopen. 14 Appendix Let −→ w = ( n, m ) ∈ ( N + ) be a weight. Lemma 5.1 Let f, g ∈ K [[ x, y ]] be power series without constant term.Then i ( f, g ) ≥ (ord −→ w ( f )) ( ord −→ w ( g )) mn , with equality if and only if the system of equations in −→ w ( f ) = 0 , in −→ w ( g ) = 0 has the only solution ( x, y ) = (0 , . Proof. By a basic property of the intersection multiplicity (see for example[19, Proposition 3.8 (v)]) we have that for any nonzero power series ˜ f , ˜ gi ( ˜ f , ˜ g ) ≥ ord( ˜ f )ord(˜ g ) , (7)with equality if and only if the system of equations in( ˜ f ) = 0, in(˜ g ) = 0has the only solution (0 , f ( u, v ) = f ( u n , v m )and ˜ g ( u, v ) = g ( u n , v m ). Then i ( ˜ f , ˜ g ) = i ( f, g ) i ( u n , v m ) = i ( f, g ) nm ,ord( ˜ f ) = ord −→ w ( f ), ord(˜ g ) = ord −→ w ( g ) and the lemma follows from (7). Lemma 5.2 Let f ∈ K [[ x, y ]] be a non-zero power series. Then i (cid:18) ∂f∂x , ∂f∂y (cid:19) ≥ (cid:18) ord −→ w ( f ) n − (cid:19) (cid:18) ord −→ w ( f ) m − (cid:19) with equality if and only if f is a semi-quasihomogeneous singularity withrespect to −→ w . Proof. The following two properties are useful:ord −→ w (cid:18) ∂f∂x (cid:19) ≥ ord −→ w ( f ) − n with equality if and only if ∂∂x in −→ w ( f ) = 0 , (8)if ∂∂x in −→ w ( f ) = 0 then in −→ w (cid:18) ∂f∂x (cid:19) = ∂∂x in −→ w ( f ) . (9)By the first part of Lemma 5.1 and Property (8) we get15 (cid:18) ∂f∂x , ∂f∂y (cid:19) ≥ (cid:16) ord −→ w (cid:16) ∂f∂x (cid:17)(cid:17) (cid:16) ord −→ w (cid:16) ∂f∂y (cid:17)(cid:17) nm ≥ (ord −→ w ( f ) − n ) ( ord −→ w ( f ) − m ) nm = (cid:18) ord −→ w ( f ) n − (cid:19) (cid:18) ord −→ w ( f ) m − (cid:19) . Using the second part of Lemma 5.1 and Properties (8) and (9) we checkthat i (cid:16) ∂f∂x , ∂f∂y (cid:17) = (cid:16) ord −→ w ( f ) n − (cid:17) (cid:16) ord −→ w ( f ) m − (cid:17) if and only if f is a semi-quasihomogeneous singularity with respect to −→ w . Lemma 5.3 (Hensel’s Lemma [13, Theorem 16.6]). Suppose that in −→ w ( f ) = ψ · · · ψ s with pairwise coprime ψ i . Then f = g · · · g s ∈ K [[ x, y ]] with in −→ w ( g i ) = ψ i for i = 1 , . . . , s . References [1] Boubakri, Y., Greuel G-M, Markwig, T.: Invariants of hypersurfacesingularities in positive characteristic. Rev. Mat. Complut. , 61-85,(2010)[2] Campillo, A.: Algebroid curves in positive characteristic. Lecture Notesin Mathematics, 813. Springer Verlag, Berlin, (1980)[3] Campillo, A.: Hamburger-Noether expansions over rings. Trans. Amer.Math. Soc. , 377-388 (1983)[4] Cassou-Nogu`es, P, P loski, A.: Invariants of plane curve singularitiesand Newton diagrams. Univ. Iag. 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