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Combinatorics of binomial primary decomposition
"... ABSTRACT. An explicit lattice point realization is provided for the primary components of an arbitrary binomial ideal in characteristic zero. This decomposition is derived from a characteristicfree combinatorial description of certain primary components of binomial ideals in affine semigroup rings, ..."
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Cited by 8 (4 self)
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ABSTRACT. An explicit lattice point realization is provided for the primary components of an arbitrary binomial ideal in characteristic zero. This decomposition is derived from a characteristicfree combinatorial description of certain primary components of binomial ideals in affine semigroup rings, namely those that are associated to faces of the semigroup. These results are intimately connected to hypergeometric differential equations in several variables. 1.
Consani Characteristic 1, entropy and the absolute point
, 1997
"... We show that the mathematical meaning of working in characteristic one is directly connected to the fields of idempotent analysis and tropical algebraic geometry and we relate this idea to the notion of the absolute point Spec F1. After introducing the notion of “perfect” semiring of characteristi ..."
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Cited by 7 (4 self)
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We show that the mathematical meaning of working in characteristic one is directly connected to the fields of idempotent analysis and tropical algebraic geometry and we relate this idea to the notion of the absolute point Spec F1. After introducing the notion of “perfect” semiring of characteristic one, we explain how to adapt the construction of the Witt ring in characteristic p> 1 to the limit case of characteristic one. This construction also unveils an interesting connection with entropy and thermodynamics, while shedding a new light on the classical Witt construction itself. We simplify our earlier construction of the geometric realization of an F1scheme and extend our earlier computations of the zeta function to cover the case of F1schemes with torsion. Then, we show that the study of the additive structures on monoids provides a natural map M ↦ → A(M) from monoids to sets which comes close to fulfill the requirements for the hypothetical curve Spec Z over the absolute point Spec F1. Finally, we test the computation of the zeta function on elliptic
Polytopal linear retractions
 Trans. Amer. Math. Soc
"... Abstract. We investigate graded retracts of polytopal algebras (essentially the homogeneous rings of affine cones over projective toric varieties) as polytopal analogues of vector spaces. In many cases we show that these retracts are again polytopal algebras and that codimension 1 retractions factor ..."
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Cited by 6 (6 self)
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Abstract. We investigate graded retracts of polytopal algebras (essentially the homogeneous rings of affine cones over projective toric varieties) as polytopal analogues of vector spaces. In many cases we show that these retracts are again polytopal algebras and that codimension 1 retractions factor through retractions preserving the semigroup structure. We expect that these results hold in general. This paper is a part of the project started in [BG1, BG2], where we have investigated the graded automorphism groups of polytopal algebras. Part of the motivation comes from the observation that there is a reasonable ‘polytopal ’ generalization of linear algebra (and, subsequently, that of algebraic Ktheory). 1.
On presentations of commutative monoids
 INT. J. ALGEBRA COMPUT
, 1999
"... In this paper, we introduce the concept of a strongly reduced monoid and we characterize the minimal presentations for such monoids. As a consequence, we give a method to obtain a presentation for any commutative monoid. ..."
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Cited by 6 (4 self)
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In this paper, we introduce the concept of a strongly reduced monoid and we characterize the minimal presentations for such monoids. As a consequence, we give a method to obtain a presentation for any commutative monoid.
C.: An algorithm for the QuillenSuslin theorem for monoid rings
 J. Pure Appl. Algebra 117
, 1997
"... Abstract. Let k be a field, and let M be a commutative, seminormal, finitely generated monoid, which is torsionfree, cancellative, and has no nontrivial units. J. Gubeladze proved that finitely generated projective modules over kM are free. This paper contains an algorithm for finding a free basis f ..."
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Cited by 5 (1 self)
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Abstract. Let k be a field, and let M be a commutative, seminormal, finitely generated monoid, which is torsionfree, cancellative, and has no nontrivial units. J. Gubeladze proved that finitely generated projective modules over kM are free. This paper contains an algorithm for finding a free basis for a finitely generated projective module over kM. As applications one obtains alternative algorithms for the QuillenSuslin Theorem for polynomial rings and Laurent polynomial rings, based on Quillen’s proof. I.
Arbitrarily Tight Bounds On The Distribution Of Smooth Integers
 Proceedings of the Millennial Conference on Number Theory
, 2002
"... This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFTbased powerseries exponentiation; explains how one can choose the parameters to achieve ..."
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Cited by 3 (1 self)
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This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFTbased powerseries exponentiation; explains how one can choose the parameters to achieve any desired level of accuracy; and discusses several generalizations.
SOME RESULTS ON NORMAL HOMOGENEOUS IDEALS
, 2002
"... Abstract. In this article we investigate when a homogeneous ideal in a graded ring is normal, that is, when all positive powers of the ideal are integrally closed. We are particularly interested in homogeneous ideals in an Ngraded ring A of the form A≥m: = ⊕ ℓ≥m Aℓ and monomial ideals in a polynomi ..."
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Cited by 3 (1 self)
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Abstract. In this article we investigate when a homogeneous ideal in a graded ring is normal, that is, when all positive powers of the ideal are integrally closed. We are particularly interested in homogeneous ideals in an Ngraded ring A of the form A≥m: = ⊕ ℓ≥m Aℓ and monomial ideals in a polynomial ring over a field. For ideals of the form A≥m we generalize a recent result of Faridi. We prove that a monomial ideal in a polynomial ring in n indeterminates over a field is normal if and only if the first n−1 positive powers of the ideal are integrally closed. We then specialize to the case of ideals of the form I(λ): = J(λ), where J(λ) = (x λ1 1,..., xλn n) ⊆ K[x1,..., xn]. To state our main result in this setting, we let ℓ = lcm(λ1,..., ̂ λi,... λn), for 1 ≤ i ≤ n, and set λ ′ = (λ1,...,λi−1, λi + ℓ, λi+1,..., λn). We prove that if I(λ ′ ) is normal then I(λ) is normal and that the converse holds with a small additional assumption. 1.
Monoids of IGtype and maximal orders
 J. ALGEBRA
, 2006
"... Let G be a finite group that acts on an abelian monoid A. If φ: A → G is a map so that φ(aφ(a)(b)) = φ(a)φ(b), for all a, b ∈ A, then the submonoid S = {(a, φ(a))  a ∈ A} of the associated semidirect product A⋊G is said to be a monoid of IGtype. If A is a finitely generated free abelian monoid o ..."
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Cited by 3 (1 self)
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Let G be a finite group that acts on an abelian monoid A. If φ: A → G is a map so that φ(aφ(a)(b)) = φ(a)φ(b), for all a, b ∈ A, then the submonoid S = {(a, φ(a))  a ∈ A} of the associated semidirect product A⋊G is said to be a monoid of IGtype. If A is a finitely generated free abelian monoid of rank n and G is a subgroup of the symmetric group Sym n of degree n, then these monoids first appeared in the work of GatevaIvanova and Van den Bergh (they are called monoids of Itype) and later in the work of Jespers and Okniński. It turns out that their associated semigroup algebras share many properties with polynomial algebras in finitely many commuting variables. In this paper we first note that finitely generated monoids S of IGtype are epimorphic images of monoids of Itype and their algebras K[S] are Noetherian and satisfy a polynomial identity. In case the group of fractions SS −1 of S also is torsionfree then it is characterized when K[S] also is a maximal order. It turns out that they often are, and hence these algebras again share arithmetical properties with natural classes of commutative algebras. The characterization is in terms of prime ideals of S, in particular Gorbits of minimal prime ideals in A play a crucial role. Hence, we first describe the prime ideals of S. It also is described when the group SS −1 is torsionfree.
Toric Varieties, Monoid Schemes and cdh descent
"... Abstract. We give conditions for the MayerVietoris property to hold for the algebraic Ktheory of blowup squares of toric varieties and schemes, using the theory of monoid schemes. These conditions are used to relate algebraic Ktheory to topological cyclic homology in characteristic p. To achieve ..."
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Cited by 2 (1 self)
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Abstract. We give conditions for the MayerVietoris property to hold for the algebraic Ktheory of blowup squares of toric varieties and schemes, using the theory of monoid schemes. These conditions are used to relate algebraic Ktheory to topological cyclic homology in characteristic p. To achieve our goals, we develop many notions for monoid schemes based on classical algebraic geometry, such as separated and proper maps and resolution of singularities. The goal of this paper is to prove Haesemeyer’s Theorem [18, 3.12] for toric schemes in any characteristic. It is proven below as Corollary 14.4. Theorem 0.1. Assume k is a commutative regular noetherian ring containing an infinite field and let G be a presheaf of spectra defined on the category of schemes of finite type over k. If G satisfies the MayerVietoris property for Zariski covers, finite abstract blowup squares, and blowups along regularly embedded closed subschemes, then G satisfies the MayerVietoris property for all abstract blowup squares of toric kschemes obtained from subdividing a fan. The application we have in mind is to understand the relationship between the