CHARACTERIZATION OF STABLY SIMPLE CURVE SINGULARITIES MUHAMMAD AHSAN BINYAMIN, JUNAID ALAM KHAN, FAIRA KANWAL JANJUA, NAVEED HUSSAIN
Abstract. In this article we characterize the classification of stably simple curve singularities given by V.I. Arnold, in terms of invariants. On the basis of this characterization we describe an implementation of a classifier for stably simple curve singularities in the computer algebra system SINGULAR.
1. Introduction A singularity of a curve is a germ of an analytic map f : (C, 0) → (Cn , 0). Bruce and Gaffney in [BG] and Gibson and Hobbs in [GH] classified simple curve singularities for the case n = 2 and n = 3 respectively. In [FP], Janjua and Pfister gave the implementation of a classifier for simple curve singularities for n ≤ 3 in the computer algebra system singular [DGPS]. In [Ar1], Arnold gave the classification of stably simple curves. The simple singularities classified by Bruce, Gaffney, Gibson and Hobbs are also stably simple. They are characterized in [FP] by invariants. In this article we characterize the remaining stably simple curve singularities f : (C, 0) → (Cn , 0), n ≥ 4 in terms of invariants and describe the implementation of a classifier for stably simple curve singularities in singular [DGPS], [GP]. A curve singularity at the origin in Cn can be described by a germ of analytic map f : (C, 0) → (Cn , 0). Let L be the group of germs of non-degenerate analytic maps (Cn , 0) → (Cn , 0) and R be the group of germs of non-degenerate analytic maps (C, 0) → (C, 0), let G = L × R. The group L is called the group of left co-ordinate changes and group R is called the group of right co-ordinate changes. Definition 1.1. The group G acts on the space of map germs f : (C, 0) → (Cn , 0) by (l, r) · f := l ◦ f ◦ r−1 , l ∈ L, r ∈ R. Two germs of curves f and g are equivalent if they lie in the same orbit under the G-action. Definition 1.2. A germ f is called simple if there exist a neighbourhood of f in the space of germs which intersects only a finite number of G-orbits. Definition 1.3. A germ f is called stably simple if it remains simple after the natural immersion (Cn , 0) ,→ (CN , 0). 2010 Mathematics Subject Classification. 14Q05,14H20. Key words and phrases. Stably equivalent, Irreducible singularities, Stably simple curve singularities, Singular. 1
2. Invariants In this section we will recall the invariants which play an important role to characterize the stably simple curve singularities. Definition 2.1. Let C[[x1 (t), x2 (t), .., xn (t)]] ⊆ C[[t]] be the sub-algebra corresponding to the germ of the curve defined by f : (C, 0) → (Cn , 0), f (t) = (x1 (t), . . . , xn (t)). Then (1) Γ := {ordt (h)|h ∈ C[[x1 (t), x2 (t), .., xn (t)]]} is called the semi-group of C[[x1 (t), x2 (t), .., xn (t)]]. (2) Let ϕ = (x(t), y(t)) be a Puiseux parametrization of a plane branch with semi group of values hv0 , v1 , . . . , vn i. Then either ϕ is A-equivalent to the monomial parametrization (tv0 , tv1 ) or it is A-equivalent to a parametrization X (tv0 , tv1 + tλ + ai ti ), λ. 3
Proposition 3.2. The type of the stably simple curve singularities listed in Table1 is completely characterized by the semi-group except in the cases: (t4 , t7 , t10 , t13 ) and (t4 , t7 + t9 , t10 , t13 ), (t4 , t6 + t2k+7 , t2k+9 , t2k+11 ) and (t4 , t6 , t2k+9 , t2k+11 ) . Proof. We will give the proof for the cases (t5 , t6 , t7 , t8 ) and (t4 , t6 +t2k+7 , t2k+9 , t2k+11 ), k ≥ 0. The proof for the other cases is similar. Case-1: Let Γ = h5, 6, 7, 8i be the semi group of C[[x1 , x2 , x3 , x4 ]]. Then we may assume that x1 = t5 , x2 = t6 + h.o.t, x3 = t7 + h.o.t, x4 = t8 + h.o.t Since the conductor is 10 then we may assume x1 = t5 , x2 = t6 + αt9 , x3 = t7 + βt9 , x4 = t8 + γt9 By using x2 = t6 + αt9 we can remove the terms βt9 and γt9 from x2 and x3 . Therefore we can assume x1 = t5 , x2 = t6 + αt9 , x3 = t7 , x4 = t8 Using the transformation t → t − α6 t4 , we may assume α = 0. In this case we obtain the singularity (t5 , t6 , t7 , t8 ). Case-2: Let Γ = h4, 6, 2k + 9, 2k + 11i be the semi group of C[[x1 , x2 , x3 , x4 ]]. Then we may assume that it corresponds to the curve x1 = t4 , x2 = t6 + h.o.t, x3 = t2k+9 + h.o.t, x4 = t2k+11 + h.o.t The conductor of semi group is 2k + 8. If x2 ∈ C[[t2 ]] then applying a suitable transformation we may assume x2 = t6 . Since the conductor of semi group is 2k + 8 we obtain after a suitable transformation the curve x1 = t4 , x2 = t6 , x3 = t2k+9 , x4 = t2k+11 . If x2 ∈ / C[[t2 ]] consider the plane curve defined by x1 and x2 . According to Hefez and Hernandes there is a transformation such that x1 = t4 and x2 = t6 + tλ , where λ is the Zariski invariant. Note that λ is odd. Considering x22 − x31 we obtain that λ + 6 ∈ Γ. This implies that λ + 6 = (2k + 9)a + (2k + 11)b + 6c + 4d for suitable a, b, c, d. Since λ is odd we have a 6= 0 or b 6= 0. This implies that λ ≥ 2k + 3, λ 6= 2k + 5. If λ = 2k + 3 then C[[x1 , x2 , x3 , x4 ]]= C[[t4 , t6 + t2k+3 , t2k+11 ]], i.e. we are in the case n = 3. If λ = 2k + 7 then C[[x1 , x2 , x3 , x4 ]]= C[[t4 , t6 + t2k+7 , t2k+9 , t2k+11 ]]. If λ ≥ 2k + 9 then C[[x1 , x2 , x3 , x4 ]]= C[[t4 , t6 , t2k+9 , t2k+11 ]]. Proposition 3.3. Let Γ = h4, 7, 10, 13i be the semi-group of (t4 + h.o.t, t7 + h.o.t, t10 + h.o.t, t13 + h.o.t). If (t4 , t7 ) is the normal form of (t4 + h.o.t, t7 + h.o.t) then (t4 , t7 , t10 , t13 ) is the normal form of (t4 +h.o.t, t7 +h.o.t, t10 +h.o.t, t13 +h.o.t). And if (t4 , t7 +t9 ) is the normal form of (t4 +h.o.t, t7 +h.o.t) then (t4 , t7 +t9 , t10 , t13 ) is the normal form of (t4 + h.o.t, t7 + h.o.t, t10 + h.o.t, t13 + h.o.t). Proof. Since by using the methods of Hefez and Hernandes we can reduce (t4 + h.o.t, t7 + h.o.t) into (t4 , t7 + tλ ), where λ is Ziriski invariant (λ = ∞ included). It is proved in [FP] that if the semi module corresponding to (t4 , t7 +tλ ) is Λ =< 3, 6 >, then the normal form of (t4 + h.o.t, t7 + h.o.t) is (t4 , t7 ). Here the semi module is the set of values of the module of differentials of C[[t4 , t7 + tλ ]]. Since the conductor of Γ = h4, 7, 10, 13i is 10, so we can assume t10 + h.o.t. → t10 and t13 + h.o.t. → t13 . 4
Hence the normal form of (t4 +h.o.t, t7 +h.o.t, t10 +h.o.t, t13 +h.o.t) is (t4 , t7 , t10 , t13 ). If semi-module corresponding to (t4 , t7 + tλ ) is Λ =< 3, 6, 16 >. Then the normal form of (t4 +h.o.t, t7 +h.o.t) is (t4 , t7 +t13 ). Then the same arguments show that the normal form of (t4 + h.o.t, t7 + h.o.t, t10 + h.o.t, t13 + h.o.t) is (t4 , t7 + t9 , t10 , t13 ). Proposition 3.4. Let Γ = h4, 6, 2k + 9, 2k + 11i be the semi-group of (t4 , t6 + h.o.t, t2k+9 + h.o.t, t2k+11 + h.o.t). If C[[t4 , t6 + h.o.t]] ⊆ C[[t2 ]] then the singularity is equivalent to (t4 , t6 , t2k+9 , t2k+11 ). If λ is the Zariski number of C[[t4 , t6 + h.o.t]] and λ = 2k + 7 then the singularity is equivalent to (t4 , t6 + t2k+7 , t2k+9 , t2k+11 ). In the other case (λ = 2k + 3) it is embedded in C3 . Proof. The proof follows from the second part of the proof of proposition-3.2.
4. The Algorithm for the Classifier We denote by classify3 the algorithm of [FP] to classify the simple space curve singularities. The following algorithm classifies the stably simple curve singularities. Algorithm 1 Stably Simple Curves (stableCur) Input: x1 (t), x2 (t), . . . , xn (t) ∈ C[[t]] such that for A = C[[x1 (t), x2 (t), . . . , xn (t)]], dimC (C[[t]]/ A) < ∞. Output: y1 (t), y2 (t), . . . , ym (t), the normal form or 0 if A is not stably simple. 1: If n ≤ 3 then return classify3 (x1 (t), x2 (t), . . . , xn (t)). 2: Compute G = {g1 , g2 , · · · , gs }, a reduced Sagbi basis of A such that LT (gi ) = tai , a1 < a2 < · · · < as . 3: Compute a minimal system z1 , z2 , . . . , zl of generators of A,ord(zi ) < ord(zi+1 ), z1 should be a monomial. 4: If l ≤ 3 then return classify3 (z1 (t), z2 (t), . . . , zl (t)). 5: Compare semi group with the list in Table-1. 6: If the semi group is Γ = h4, 7, 10, 13i. Then compute classify3 (z1 (t), z2 (t)). If classify3 (z1 (t), z2 (t)) = (t4 , t7 ) then return (t4 , t7 , t10 , t13 ). If classify3 (z1 (t), z2 (t)) = (t4 , t7 + t13 ) then return (t4 , t7 + t13 , t10 , t13 ). 7: If the semi group is h4, 6, 2k + 9, 2k + 11i. If C[[z1 , z2 ]] ⊆ C[[t2 ]] then return (t4 , t6 , t2k+9 , t2k+11 ). return (t4 , t6 + t2k+7 , t2k+9 , t2k+11 ). 8: If the semi group is in the list then return the corresponding normal form. 9: return(0). Remark 4.1. We have implemented Algorithm-1 in the computer algebra system SINGULAR [DGPS]. Code can be download from mathcity.org/junaid. Acknowledgements: We would like to thank Dr. Gerhard Pfister for useful conversations.
References [Ar1] Arnold, V.I.: Simple singularities of curves. (Russia) Tr. Mat. Inst. Steklova 226 (1999), Mat. Fiz. probl. Kvantovoi Teor. Polya, 27-35; translation in Proc. Steklov Inst. Math. 226 (1999), 20-28. 5
[BG] Bruce,J.W; Gaffney,T.J.:Simple Singularities of Mappings C, 0 → C2 , 0. J.London Math.Soc.(2)(1982),465-474. [DGPS] Decker, W.; Greuel, G.-M.; Pfister, G.; Sch¨ onemann, H.: Singular 3-1-6 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2013). [FP] Janjua, F.K; Pfister, G.: A classifier for simple space curve singularities. To appear in Studia Scientiarum Mathematicarum Hungarica 2014. [GH] Gibson,C.G; Hobbs,C.A.:Simple Singularities of Space Curves. Mathematics. Proc. Camb. Phil. Soc.(1993),113,297. [GP] Greuel, G.-M.; Pfister, G.: A Singular Introduction to Commutative Algebra. Second edition, Springer (2007). [HH] Hefez,A;Hernandes,M.E.:Standard bases for local rings of branches and their modules of differentials.Journal of Symbolic Computation 42(2007) 178-191. [HHA] Hefez,A;Hernandes,M.E.:The Analytic Classification Of Plane Branches. Bulletin of the London Mathematical Society,Volume 43, Number 2, 26 April 26,(2011), pp.289-298(10). Muhammad Ahsan Binyamin, Department of Mathematics, GC University, Faisalabad, Pakistan E-mail address:
[email protected] Junaid Alam Khan, Department of Mathematical Sciences, Institute of Business Administration, Karachi, Pakistan E-mail address:
[email protected] Faira Kanwal Janjua, Institute of Business and Management, University of Engineering and Technology, Lahore, Pakistan E-mail address:
[email protected] Naveed Hussain, Department of Mathematics, GC University, Faisalabad, Pakistan E-mail address:
[email protected]
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