11,532 research outputs found
Binomial Difference Ideal and Toric Difference Variety
In this paper, the concepts of binomial difference ideals and toric
difference varieties are defined and their properties are proved. Two canonical
representations for Laurent binomial difference ideals are given using the
reduced Groebner basis of Z[x]-lattices and regular and coherent difference
ascending chains, respectively. Criteria for a Laurent binomial difference
ideal to be reflexive, prime, well-mixed, perfect, and toric are given in terms
of their support lattices which are Z[x]-lattices. The reflexive, well-mixed,
and perfect closures of a Laurent binomial difference ideal are shown to be
binomial. Four equivalent definitions for toric difference varieties are
presented. Finally, algorithms are given to check whether a given Laurent
binomial difference ideal I is reflexive, prime, well-mixed, perfect, or toric,
and in the negative case, to compute the reflexive, well-mixed, and perfect
closures of I. An algorithm is given to decompose a finitely generated perfect
binomial difference ideal as the intersection of reflexive prime binomial
difference ideals.Comment: 72 page
Local analysis of Grauert-Remmert-type normalization algorithms
Normalization is a fundamental ring-theoretic operation; geometrically it
resolves singularities in codimension one. Existing algorithmic methods for
computing the normalization rely on a common recipe: successively enlarge the
given ring in form an endomorphism ring of a certain (fractional) ideal until
the process becomes stationary. While Vasconcelos' method uses the dual
Jacobian ideal, Grauert-Remmert-type algorithms rely on so-called test ideals.
For algebraic varieties, one can apply such normalization algorithms
globally, locally, or formal analytically at all points of the variety. In this
paper, we relate the number of iterations for global Grauert-Remmert-type
normalization algorithms to that of its local descendants.
We complement our results by an explicit study of ADE singularities. This
includes the description of the normalization process in terms of value
semigroups of curves. It turns out that the intermediate steps produce only ADE
singularities and simple space curve singularities from the list of
Fruehbis-Krueger.Comment: 22 pages, 7 figure
Computing the canonical representation of constructible sets
Constructible sets are needed in many algorithms of Computer Algebra, particularly in the GröbnerCover and other algorithms for parametric polynomial systems. In this paper we review the canonical form ofconstructible sets and give algorithms for computing it.Peer ReviewedPostprint (author's final draft
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