Computing the degree of a lattice ideal of dimension one

Abstract. We study the regularity and the algebraic properties of certain lattice ideals. We establish a map I → I between the family of graded lattice ideals in an N-graded polynomial ring over a field K and the family of graded lattice ideals in a polynomial ring with the standard grading. This map is shown to preserve the complete intersection property and the regularity of I but not the degree. We relate the Hilbert series and the generators of I and I. If dim(I) = 1, we relate the degrees of I and I. It is shown that the regularity of certain lattice ideals is additive in a certain sense. Then, we give some applications. For finite fields, we give a formula for the regularity of the vanishing ideal of a degenerate torus in terms of the Frobenius number of a semigroup. We construct vanishing ideals, over finite fields, with prescribed regularity and degree of a certain type. Let X be a subset of a projective space over a field K. It is shown that the vanishing ideal of X is a lattice ideal of dimension 1 if and only if X is a finite subgroup of a projective torus. For finite fields, it is shown that X is a subgroup of a projective torus if and only if X is parameterized by monomials. We express the regularity of the vanishing ideal over a bipartite graph in terms of the regularities of the vanishing ideals of the blocks of the graph.

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“…, d s /r). As I and I are graded lattice ideals of dimension 1, according to some results of [27] and [20]…”

Section: Lattice Ideals Of Dimensionmentioning

confidence: 99%

Regularity and algebraic properties of certain lattice ideals

Neves

Pinto

Villarreal

2014

Bull Braz Math Soc, New Series

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“…Remark 5.8. There is an overlap between the results of Section 5 and the results of [LV,Section 3]. (Recall that an integer matrix and its transpose have the same invariant factors.)…”

Section: A Review Of the Normal Decomposition Of An Integer Matrixmentioning

confidence: 99%

The primary components of positive critical binomial ideals

O’Carroll

Planas-Vilanova

2013

Journal of Algebra

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A natural candidate for a generating set of the (necessarily prime) defining ideal of an n-dimensional monomial curve, when the ideal is an almost complete intersection, is a full set of n critical binomials. In a somewhat modified and more tractable context, we prove that, when the exponents are all positive, critical binomial ideals in our sense are not even unmixed for n ≥ 4, whereas for n ≤ 3 they are unmixed. We further give a complete description of their isolated primary components as the defining ideals of monomial curves with coefficients. This answers an open question on the number of primary components of Herzog-Northcott ideals, which comprise the case n = 3. Moreover, we find an explicit, concrete description of the irredundant embedded component (for n ≥ 4) and characterize when the hull of the ideal, i.e., the intersection of its isolated primary components, is prime. Note that these last results are independent of the characteristic of the ground field. Our techniques involve the Eisenbud-Sturmfels theory of binomial ideals and Laurent polynomial rings, together with theory of Smith Normal Form and of Fitting ideals. This gives a more transparent and completely general approach, replacing the theory of multiplicities used previously to treat the particular case n = 3.Date: September 15, 2018. 2000 Mathematics Subject Classification. 13A05, 13A15 (primary), 13P05, 13P15 (secondary).

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“…The index of regularity of S/I(X), denoted by reg(S/I(X)), is the least integer r ≥ 0 such that h X (d) = H X (d) for d ≥ r. The degree and the Krull dimension are denoted by deg(S/I(X)) and dim(S/I(X)), respectively. [12], [24]) If X is a finite set and r = reg(S/I(X)), then…”

Section: Preliminariesmentioning

confidence: 99%

Direct products in projective Segre codes

Tochimani

Pinto

Villarreal

2016

Finite Fields and Their Applications

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Let K = Fq be a finite field. We introduce a family of projective Reed-Muller-type codes called projective Segre codes. Using commutative algebra and linear algebra methods, we study their basic parameters and show that they are direct products of projective Reed-Mullertype codes. As a consequence we recover some results on projective Reed-Muller-type codes over the Segre variety and over projective tori.

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Computing the degree of a lattice ideal of dimension one

Cited by 8 publications

References 39 publications

Regularity and algebraic properties of certain lattice ideals

Regularity and algebraic properties of certain lattice ideals

The primary components of positive critical binomial ideals

Direct products in projective Segre codes

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