Computing zero-dimensional schemes

Abbott, John; Kreuzer, Martin; Robbiano, Lorenzo

doi:10.1016/j.jsc.2004.09.001

Cited by 13 publications

(21 citation statements)

References 9 publications

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“…See Theorem 6.2.12 in [9] for a proof. 3,12). As the elements in π 1 , π 2 are pairwise distinct, we can make the distraction of I with respect to π = (π 1 , π 2 ):…”

Section: For Every Power Product T = X α1mentioning

confidence: 99%

“…Furthermore the vanishing ideal I(Y) of a finite set Y of s points is a zerodimensional radical ideal in P of type I(Y) = m 1 ∩ · · · ∩ m s , and which we also call an ideal of points. For an introduction to ideals of points, see [9], Section 6.3; for methods to efficiently compute them and other zero-dimensional ideals, see [2] and [3].…”

Section: Introductionmentioning

confidence: 99%

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Small Gröbner fans of ideals of points

Dimitrova

Robbiano

et al. 2019

J. Algebra Appl.

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In the context of modeling biological systems, it is of interest to generate ideals of points with a unique reduced Gröbner basis, and the first main goal of this paper is to identify classes of ideals in polynomial rings which share this property. Moreover, we provide methodologies for constructing such ideals. We then relax the condition of uniqueness. The second and most relevant topic discussed here is to consider and identify pairs of ideals with the same number of reduced Gröbner bases, that is, with the same cardinality of their associated Gröbner fan.the space of PDSs that fit the data and a minimal model is selected from the space by computing a reduced Gröbner basis of the ideal and taking the normal forms of the model equations. While this provides an algorithmic solution to model selection, each choice of monomial order results in a different minimal PDS, with each one yielding different hypotheses about the underlying biological network. The following example illustrates this claim.Lactose metabolism in E.coli is controlled by the lac operon, a genetic system made up of simultaneously transcribed genes. It is said that the lac operon (x) is ON (lactose is metabolized) when the activating protein CAP (y) is present and when the inhibiting protein lacI (z) is absent. This behavior can be described by the Boolean function f = y ∧ ¬z; as a polynomial over the finite field F 2 , we can write f = y(z + 1) = yz + y. If we consider the inputs X = {(1, 0, 0), (0, 1, 0), (1, 0, 1)} representing Boolean states for the lac operon, CAP, and lacI respectively, then the ideal of polynomials vanishing on X has two Gröbner bases, namely {x 2 + x, z 2 + z, y + x + 1, xz + z} and {y 2 + y, z 2

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“…See Theorem 6.2.12 in [9] for a proof. 3,12). As the elements in π 1 , π 2 are pairwise distinct, we can make the distraction of I with respect to π = (π 1 , π 2 ):…”

Section: For Every Power Product T = X α1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Small Gröbner fans of ideals of points

Dimitrova

Robbiano

et al. 2019

J. Algebra Appl.

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“…The following easy properties of the shifted-homogenization will help the reader understand the proof of Theorem 5.3 Lemma 2. 6. Let P be a polynomial ring over the field K, and let f, g ∈ P .…”

Section: Notation and Terminologymentioning

confidence: 99%

Implicitization of hypersurfaces

Abbott

Bigatti

Robbiano

2017

Journal of Symbolic Computation

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We present new, practical algorithms for the hypersurface implicitization problem: namely, given a parametric description (in terms of polynomials or rational functions) of the hypersurface, find its implicit equation. Two of them are for polynomial parametrizations: one algorithm, "ElimTH", has as main step the computation of an elimination ideal via a truncated, homogeneous Gröbner basis. The other algorithm, "Direct", computes the implicitization directly using an approach inspired by the generalized Buchberger-Möller algorithm. Either may be used inside the third algorithm, "RatPar", to deal with parametrizations by rational functions. Finally we show how these algorithms can be used in a modular approach, algorithm "ModImplicit", for avoiding the high costs of arithmetic with rational numbers. We exhibit experimental timings to show the practical efficiency of our new algorithms.

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“…Also the trick of intersecting with the hyperplane {X 1 = 1} for passing from projective to affine points is being used. See [ABKR00] and [AKR05] for details on the stopping criterion and the projective-to-affine trick. A version of the interpolation technique of Section 4 already appear in the author's paper [Led08].…”

Section: Final Remarksmentioning

confidence: 99%

Finite sets of d-planes in affine space

Lederer

2009

Journal of Algebra

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Let A be a subvariety of affine space A n whose irreducible components are d-dimensional linear or affine subspaces of A n . Denote by D( A) ⊂ N n the set of exponents of standard monomials of A. We show that the combinatorial object D( A) reflects the geometry of A in a very direct way. More precisely, we define a d-plane in N n as being a set γ + j∈ J Ne j , where # J = d and γ j = 0 for all j ∈ J . We call the d-plane thus defined to be parallel to j∈ J Ne j . We show that the number of d-planes in D( A) equals the number of components of A. This generalises a classical result, the finiteness algorithm, which holds in the case d = 0. In addition to that, we determine the number of all d-planes in D( A) parallel to j∈ J Ne j , for all J . Furthermore, we describe D( A) in terms of the standard sets of the intersections A ∩ {X 1 = λ}, where λ runs through A 1 .

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Computing zero-dimensional schemes

Cited by 13 publications

References 9 publications

Small Gröbner fans of ideals of points

Small Gröbner fans of ideals of points

Implicitization of hypersurfaces

Finite sets of d-planes in affine space

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