On Fulton’s Algorithm for Computing Intersection Multiplicities

Marcus, Steffen; Maza, Marc Moreno; Vrbik, Paul

doi:10.1007/978-3-642-32973-9_17

Cited by 9 publications

(10 citation statements)

References 13 publications

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“…This representation is well-understood and implemented since already more than two decades, especially for radical ideals. Although several attempts to represent multiplicities have come out [7,27,29,1] they are limited in scope, and resort to sophisticated concepts while obtaining partial information only. Triangular sets, hence standard triangular decomposition methods, cannot in general produce an ideal preserving decomposition: for example a mere primary ideal in dimension zero is not triangular in general: (think of the primary ideal y 2 + 3x, xy + 2x, x 2 of associated prime x, y ; it is the reduced lexGb for x ≺ y and not radical).…”

Section: Context and Resultsmentioning

confidence: 99%

Lexicographic Groebner bases of bivariate polynomials modulo a univariate one

Dahan

2020

Preprint

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Let T (x) ∈ k[x] be a monic non-constant polynomial and write R = k[x]/ T the quotient ring. Consider two bivariate polynomials a(x, y), b(x, y) ∈ R [y]. In a first part, T = p e is assumed to be the power of an irreducible polynomial p. A new algorithm that computes a minimal lexicographic Gröbner basis of the ideal a, b, p e , is introduced. A second part extends this algorithm when T is general through the "local/global" principle realized by a generalization of "dynamic evaluation", restricted so far to a polynomial T that is squarefree. The algorithm produces splittings according to the case distinction "invertible/nilpotent", extending the usual "invertible/zero" in classic dynamic evaluation. This algorithm belongs to the Euclidean family, the core being a subresultant sequence of a and b modulo T . In particular no factorization or Gröbner basis computations are necessary. The theoretical background relies on the Lazard's structural theorem for lexicographic Gröbner bases in two variables. An implementation is realized in Magma, and extensive benchmarks are provided that show clearly the benefits, sometimes huge, of this approach compared to the computation of Gröbner bases.

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Section: Context and Resultsmentioning

confidence: 99%

Lexicographic Groebner bases of bivariate polynomials modulo a univariate one

Dahan

2020

Preprint

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“…But none provided a structural isomorphism that mimics the case of gcd of polynomials over a field. In a different direction, some works have focused on representing not only solution points, but their multiplicity as well [3,9,12,1]. The multiplicity is a much coarser information than what provides an ideal isomorphism; Moreover the methods proposed in these works are not simpler than the one proposed in [4] and here.…”

Section: Related Workmentioning

confidence: 99%

Computation of gcd chain over the power of an irreducible polynomial

Dahan

2018

Preprint

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A notion of gcd chain has been introduced by the author at ISSAC 2017 for two univariate monic polynomials with coefficients in a ring R = k[x1, . . . , xn]/ T where T is a primary triangular set of dimension zero. A complete algorithm to compute such a gcd chain remains challenging. This work treats completely the case of a triangular set T = (T1(x)) in one variable, namely a power of an irreducible polynomial. This seemingly "easy" case reveals the main steps necessary for treating the general case, and it allows to isolate the particular one step that does not directly extend and requires more care.

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“…Following that strategy, three of the co-authors of this note have proposed, in another recent work [8], an extension of Fulton's algorithm for computing the intersection multiplicity of two plane curves at any of their intersection points. Indeed, as pointed out by Fulton in his Intersection Theory, the intersection multiplicity of two plane curves V (f ) and V (g) satisfy a series of seven properties which uniquely define I(p; f, g) at each point p ∈ V (f, g).…”

Section: Overviewmentioning

confidence: 99%

“…, x n . In [8], two extensions of Fulton's algorithm are proposed. First, thanks to the regularity test for regular chains, the construction is adapted such that it can work correctly at any point of V (f, g), rational or not.…”

Section: Overviewmentioning

confidence: 99%

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Doing algebraic geometry with the RegularChains library

Maza

2015

ACM Commun. Comput. Algebra

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Abstract. Traditionally, Groebner bases and cylindrical algebraic decomposition are the fundamental tools of computational algebraic geometry. Recent progress in the theory of regular chains has exhibited efficient algorithms for doing local analysis on algebraic varieties. In this note, we present the implementation of these new ideas within the module AlgebraicGeometryTools of the RegularChains library. The functionalities of this new module include the computation of the (non-trivial) limit points of the quasi-component of a regular chain. This type of calculation has several applications like computing the Zarisky closure of a constructible set as well as computing tangent cones of space curves, thus providing an alternative to the standard approaches based on Groebner bases and standard bases, respectively. From there, we have derived an algorithm which, under genericity assumptions, computes the intersection multiplicity of a zero-dimensional variety at any of its points. This algorithm relies only on the manipulations of regular chains.

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On Fulton’s Algorithm for Computing Intersection Multiplicities

Cited by 9 publications

References 13 publications

Lexicographic Groebner bases of bivariate polynomials modulo a univariate one

Lexicographic Groebner bases of bivariate polynomials modulo a univariate one

Computation of gcd chain over the power of an irreducible polynomial

Doing algebraic geometry with the RegularChains library

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