Mullhaupt, Ph.

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Previous Up Next ArticleCitationsFrom References: 0From Reviews: 0MR2284291 (Review) 13N05 (13F25 13P10)Hefez, A. (BR-UFFM); Hernandes, M. E. (BR-MAR)Standard bases for local rings of branches and their modules of differentials. (Englishsummary)J. Symbolic Comput. 42 (2007), no. 1-2, 178–191.A topological and analytical classification of planar branches introduces some invariant numbersassociated with a given planar branch that distinguishes the type of singularity of the branch. Forexample, a planar branch in the ordinary plane is cut out by two formal power series in x and y,and one attempt to classify the branch is by performing successive coordinate transformations towrite the resulting expression in a normal form in which the type and singularity can be readilyread off. Abstracting a little further, as is now widely done in algebraic geometry, a branch overan algebraically closed field K is a prime ideal C = 〈f1, f2, . . . , fr〉 of the ring of formal powerseries K[[X1, X2, . . . , Xn]] such that its local ring has Krull dimension one. Then the question ishow to provide the algebraic classification and the singularity type of this local ring.The paper proposes a computational technique that is based on the extension of classical poly-nomial ring theory over a field to more general rings, i.e., the generalization of algorithms such asthe Buchberger division process and the Gro¨bner basis construction. The main difference here isthat the local ring is not necessarily generated over a field and this accounts for some difficultiesin the stopping condition for the construction of the basis. These bases are called standard bases.The singularity type of the branch is closely linked to a specific valuation on the implicitizationof the branch. For instance, x2 = y3 admits x = t3 and y = t2 as an implicit parametrization,and the valuation is then given by the degree of t. The standard basis algorithm proposed gives aset of formal power series representations associated with the branch. The monomials of highestorder of each member of the standard basis reveal the type of singularity. The conductor—an important topological invariant of the branch—is the lowest number c of the semigroup ofvaluation Γ. This semigroup is characterized by the lowest degree of t in the components of theimplicitization of the branch. Whenever an element of the local ideal of the branch has valuationc1 ≥ c, then necessarily there exists another element in the ideal with valuation c1+1. The mainproposition is the following:G= {h0, . . . , hg} ⊂ O is a minimal standard basis for O if, and onlyif, {v(h0), . . . , v(hg)} is a minimal set of generators of the semigroup of values Γ of O. (Here,v(·) stands for the valuation in t.) Two irreducible algebroid curves are said to be equisingular ifthey have the same semigroup of values Γ.As an example for computing Γ, consider the branch given in implicit form x = t8, y = t10+t13 and z = t12 + at15. Since 8, 10, 12 ∈ Γ and 27 = v(x3 − z2) ∈ Γ if a 6= 0 or 23 = v(y2 −xz) ∈ Γ if a = 0, it is simple to verify that the consecutive integers 54, 55, . . . , 61 belong to Γ.Hence the conductor c≤ 54. The minimal S-processes (i.e., analogs of S-polynomials in classicalGro¨bner bases) obtained by the proposed algorithm are y4 − x5, y2 − xz, z2 − x3, y2z − x4,z3− x2y2 and z4− x6. Then, reducing modulo x, y, z using the implicitization, it is shown thatF1 = {t8, t10+ t13, t12, 2t23+ t26} so that the set of degrees of the leading monomials becomesthe generators of Γ, i.e., Γ = 〈8, 10, 12, 23〉. Now, adding elements of Γ together gives rise tothe sequence 8, 10, 12, 16, 18, 20, 22, 23, 24, 26, 28, 30, 31, 32, 34, 35, . . . , that is, all consecutiveintegers beyond 34 are generated through such combinations, so that the conductor is c = 34,which is smaller than 54.Similar constructions are obtained for the module of Ka¨hler differentials, giving a powerfulprocedure for computing Milnor’s µ and Tyurina’s τ numbers (analytical invariants). Let Λ =v(OdO/T), where T is the torsion submodule of the module of differentials OdO.Using the relation(1) τ = l(T) = µ−#(ΛrΓ),where # indicates the number of elements in the set, together with its relation with the conductor,µ = c, the computational procedures proposed can be used to compute τ . For example, considerC as the curve parameterized by x = t8, y = t12+ t13 whose associated semigroup of values isΓ = 〈8, 12, 25〉 with conductor c = 80. Computing the differentials dx, dy, and dz where z =y2−x3, the proposed standard basis algorithm provides: ω1 = 3ydx−2xdy, ω2 = 8xdz−25zdx,ω3 = 12ydz− 25zdy, and ω4 as the reduction of 25x2zdx− 8y2dz by 202/25zdz and ω5 as thereduction of yzdx+ 8x3ω1 by 6zω1. (The reduction resorts to a dedicated algorithm that goesalong with the one used for the basis construction.) Therefore, after implicitization, ψ(ω1) =−2t20, ψ(ω2) = 8t33, ψ(ω3) =−38t37− 13t38, ψ(ω4) = 20425 t50+ 5225t51 and ψ(ω5) =−4t46, fromwhichτ = c−#(ΛrΓ) =80−#{21, 29, 34, 38, 42, 46, 47, 51, 54, 55, 59, 63, 67, 71, 79}= 65.The paper gives all the details on the above ideas with complete descriptions of algorithmsand proofs. It shows how existing identities such as (1) can be put to good use so as to guidethe algorithm in obtaining a standard basis. Such algorithms are useful in various situations forestablishing key topological and analytical invariants.Reviewed by Philippe A. Mu¨llhauptReferences1. 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