Problems and algorithms for affine semigroups

Bruns, Winfried; Gubeladze, Joseph; Trung, Ngô Viêt

doi:10.1007/s002330010099

Cited by 30 publications

(21 citation statements)

References 41 publications

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“…Clearly, for a given d we have c ncm d 1 . In (the proof of) Theorem 5.3 we proved as d → ∞ the ratio d 1 /c → 1 (hence a fortiori the ratio ncm/c → 1).…”

Section: Numerical Resultsmentioning

confidence: 99%

“…. , a k ) = 1 ensures that there are elements of Γ in each congruence class mod d. The generating set {(d, 0), (d − a 1 , a 1 ), (d − a 2 , a 2 ) · · · (d − a k−1 , a k−1 ), (0, d)} is the Hilbert basis Hilb(S) of S in the language of [1]. The scheme Proj(R S ) is a projective monomial curve of degree d whose homogeneous coordinate ring is R S .…”

Section: Introductionmentioning

confidence: 99%

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Non-Cohen–Macaulay projective monomial curves

Reid

Roberts

2005

Journal of Algebra

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“…Clearly, for a given d we have c ncm d 1 . In (the proof of) Theorem 5.3 we proved as d → ∞ the ratio d 1 /c → 1 (hence a fortiori the ratio ncm/c → 1).…”

Section: Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Non-Cohen–Macaulay projective monomial curves

Reid

Roberts

2005

Journal of Algebra

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“…The basic tool is a characterization of the complement of a monomial ideal (see [12]). Some of the arguments are of interest for the study of affine semigroups and toric varieties [9,31].…”

Section: Theorem 2 (I) There Exists a Monomial Idealmentioning

confidence: 99%

“…, ν m ) ∈ Γ and we can replace ν j by ν j . After at most m such replacements we will construct a point satisfying (9).…”

Section: Claimmentioning

confidence: 99%

Integer Programs with Prescribed Number of Solutions and a Weighted Version of Doignon-Bell-Scarf’s Theorem

Aliev

Loera

Louveaux

2014

Integer Programming and Combinatorial Optimization

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Abstract. In this paper we study a generalization of the classical feasibility problem in integer linear programming, where an ILP needs to have a prescribed number of solutions to be considered solved. We first provide a generalization of the famous Doignon-Bell-Scarf theorem: Given an integer k, we prove that there exists a constant c(k, n), depending only on the dimension n and k, such that if a polyhedron {x : Ax ≤ b} contains exactly k integer solutions, then there exists a subset of the rows of cardinality no more than c(k, n), defining a polyhedron that contains exactly the same k integer solutions. The second contribution of the article presents a structure theory that characterizes precisely the set Sg ≥k (A) of all vectors b such that the problem Ax = b, x ≥ 0, x ∈ Z n , has at least k-solutions. We demonstrate that this set is finitely generated, a union of translated copies of a semigroup which can be computed explicitly via Hilbert bases computation. Similar results can be derived for those right-hand-side vectors that have exactly k solutions or fewer than k solutions. Finally we show that, when n, k are fixed natural numbers, one can compute in polynomial time an encoding of Sg ≥k (A) as a generating function, using a short sum of rational functions. As a consequence, one can identify all right-hand-side vectors that have exactly k solutions (similarly for at least k or less than k solutions). Under the same assumptions we prove that the k-Frobenius number can be computed in polynomial time.

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“…. , (d − a k−1 , a k−1 ), (0, d)} is the Hilbert basis Hilb(S) of S in the language of [1]. The scheme C S = Proj(R S ) is a projective monomial curve of degree d whose homogeneous coordinate ring is R S .…”

Section: Introductionmentioning

confidence: 99%

Maximal and Cohen–Macaulay projective monomial curves

Reid

Roberts

2007

Journal of Algebra

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In this paper we continue the investigation of Cohen-Macaulay projective monomial curves begun in [Les Reid, Leslie G. Roberts, Non-Cohen-Macaulay projective monomial curves, J. Algebra 291 (2005) 171-186]. In the process we introduce maximal curves. Cohen-Macaulay curves are maximal, but not conversely. We show that the number of all curves of degree d that are Cohen-Macaulay grows exponentially, but not as fast as the total number of curves, and also that maximal curves of degree d with sufficiently large embedding dimension relative to d are Cohen-Macaulay.

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Problems and algorithms for affine semigroups

Cited by 30 publications

References 41 publications

Non-Cohen–Macaulay projective monomial curves

Non-Cohen–Macaulay projective monomial curves

Integer Programs with Prescribed Number of Solutions and a Weighted Version of Doignon-Bell-Scarf’s Theorem

Maximal and Cohen–Macaulay projective monomial curves

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