### Local decomposition algorithms [chapter]

Maria Emilia Alonso, Teo Mora, Mario Raimondo
1991 Lecture Notes in Computer Science
Univ. Complutense Madrid Teo Mora -Univ. Genova Mario Raimondo -Univ. Genova For an ideal I ⊂ P := k[X 1 ,...,X n ], many algorithms using Gröbner techniques are a direct consequence of the definition itself of Gröbner basis: among them we can list algorithms for computing syzygies, dimension, minimal bases, free resolutions, Hilbert function and other algebro-geometric invariants of an ideal; the explicit knowledge of syzygies allows moreover to compute ideal intersection and quotients. Other
more » ... lgorithms require a specific property of Gröbner bases w.r.t. some special orderings (elimination orderings, i.e. orderings s.t. if m ∈ and n < m then n ∈ ): by computing a Gröbner basis of I w.r.t. such an ordering, one gets Gröbner bases for I d := I ∩ k[X 1 ,...,X d ] and, if I d = (0), for Iּk(X 1 ,...,X d )[X d+1 ,...,X n ]; thus, one obtains effective elimination and, as a consequence, algorithms for the computation of radicals, primary decomposition, equidimensional decomposition; primality, radicality, equidimensionality tests; solutions of systems of equations, i.e. description of the variety of zeroes (e.g. by giving the primary decomposition of the ideal, and, for each primary J, describing the associate irreducible component by giving the generic zero of Rad(J) in an explicit field extension of k); elimination provides also tests for dimension, ideal intersection and quotients. Such decomposition algorithms, together with other algorithms based on elimination, have been introduced in [GTZ]; while the algorithms in [GTZ] have some weaknesses, which we have remarked in the paper, no alternative to them is known at this moment: an alternative has been proposed by [BGS], but we don't have sufficient information on it, so we could not use it. In contrast much research has been recently devoted to radical computation: [KR] and [KL] use schemes similar to the one of [GTZ], [EH] applies cohomological techniques and the Jacobian ideal, [GH] applies ideal quotients. We propose here a new algorithm for radical computation; it is a variant of the algorithm in [GH] which doesn't require generic position. Instead of considering "global" varieties, i.e. varieties defined in an affine or projective space, one can study "local" varieties, by considering appropriate local rings: a) the localization of the polynomial ring at the origin: description of a variety near a point b) the ring of (convergent -algebraic) formal power series: analytical branches of a variety near a point c) the localization at a prime ideal of a coordinate ring: description of a variety "near a subvariety" Standard bases in local rings are a direct generalization of Gröbner bases in the sense that the definition and the basic properties generalize the corresponding definition and properties of Gröbner bases. The tangent cone algorithm which works in k[X,Y] 1+(X) is a generalization-modification of Buchberger algorithm which is based on these properties and which by suitable representation techniques can be used to compute standard bases in all the three cases listed above [MPT]. As a consequence, on a "representable" local ring many problems can be solved in essentially the same way as on polynomial rings; this allows the computation of algebro-geometric invariants (syzygies, dimension, free resolutions, Hilbert function, regularity, system of parameters) and of ideal intersections and quotients [MOR]. The elimination properties of Gröbner bases are however not directly generalizable. The main reason for that is a practical one: standard basis computation requires orderings s.t. X i < 1 for each i and then one has, say, XY < Y < X < 1, so that elimination orderings cannot be used for standard basis computations. The basic case of effective "local elimination" is the following: given a basis F of an ideal I in B : to compute a basis of Jּ:=ּI ∩ A where A := k[X] 1+(X) , and, if J = 0, of Iּk(X)[Y] 1+(Y). A recent generalization of the tangent cone algorithm [MPT] allows to compute standard bases in A[Y] w.r.t. to an ordering s.t. X i < 1 < Y j ∀i,j and if m ∈ and n < m then n ∈ . Since w.l.o.g. F ⊂ A[Y], if L is the ideal in A[Y] generated by F, then L ∩ A = I ∩ A = J; moreover, if J = (0), Lּk(X)[Y] 1+(Y) = Iּk(X)[Y] 1+(Y) . So by the generalized tangent cone algorithm, the above local elimination problem is solvable. As a direct consequence one gets elimination algorithms for localizations at the origin and (by essentially using the Weierstrass and Noether algorithms in [AMR]) for rings of algebraic series w.r.t. generic coordinates. As a consequence, the whole host of elimination-based algorithms is at least in principle available in the local case too. Most of the paper is devoted to show how the general scheme for decomposition algorithms in polynomial rings introduced in [GTZ] can be generalized to the local case. This scheme allows equidimensional decomposition and equidimensionality tests, and reduces the problems of primary decomposition, radical computation, radicality test, primality and primariety tests to (resp.) factorization, squarefree algorithm, squarefree test, irreducibility test for univariate polynomials over algebraic extensions of a suitable field, which is a trascendental extension of k in the case of polynomial rings,