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David J. Grynkiewicz - One of the best experts on this subject based on the ideXlab platform.
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Kemperman’s Critical Pair Theory
Structural Additive Theory, 2013Co-Authors: David J. GrynkiewiczAbstract:The goal of this chapter is to prove one of the most important inverse results for an arbitrary abelian group—the Kemperman Structure Theorem (KST)—which determines all Finite, Nonempty subsets A, B⊆G with |A+B|≤|A|+|B|−1. As the proof is already involved enough for the case when A and B are Finite, we only present the result in this case.
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Properties of two-dimensional sets with small sumset
Journal of Combinatorial Theory Series A, 2010Co-Authors: David J. Grynkiewicz, Oriol SerraAbstract:We give tight lower bounds on the cardinality of the sumset of two Finite, Nonempty subsets A,B@?R^2 in terms of the minimum number h"1(A,B) of parallel lines covering each of A and B. We show that, if h"1(A,B)>=s and |A|>=|B|>=2s^2-3s+2, then|A+B|>=|A|+(3-2s)|B|-2s+1. More precise estimations are given under different assumptions on |A| and |B|. This extends the 2-dimensional case of the Freiman 2^d-Theorem to distinct sets A and B, and, in the symmetric case A=B, improves the best prior known bound for |A|=|B| (due to Stanchescu, and which was cubic in s) to an exact value. As part of the proof, we give general lower bounds for two-dimensional subsets that improve the two-dimensional case of estimates of Green and Tao and of Gardner and Gronchi, related to the Brunn-Minkowski Theorem.
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Long Arithmetic Progressions in Sets with Small Sumset
arXiv: Number Theory, 2009Co-Authors: Itziar Bardaji, David J. GrynkiewiczAbstract:Let $A, B\subseteq \mathbb{Z}$ be Finite, Nonempty subsets with $\min A=\min B=0$, and let $$\delta(A,B)={\begin{array}{ll} 1 & \hbox{if} A\subseteq B, 0 & \hbox{otherwise.} If $\max B\leq \max A\leq |A|+|B|-3$ and \label{one}|A+B|\leq |A|+2|B|-3-\delta(A,B), then we show $A+B$ contains an arithmetic progression with difference 1 and length $|A|+|B|-1$. As a corollary, if \eqref{one} holds, $\max(B)\leq \max(A)$ and either $\gcd(A)=1$ or else $\gcd(A+B)=1$ and $|A+B|\leq 2|A|+|B|-3$, then $A+B$ contains an arithmetic progression with difference 1 and length $|A|+|B|-1$.
Lilu Zhao - One of the best experts on this subject based on the ideXlab platform.
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On restricted sumsets over a field
arXiv: Number Theory, 2014Co-Authors: Lilu ZhaoAbstract:We consider restricted sumsets over field $F$. Let\begin{align*}C=\{a_1+\cdots+a_n:a_1\in A_1,\ldots,a_n\in A_n, a_i-a_j\notin S_{ij}\ \text{if}\ i\not=j\},\end{align*} where $S_{ij}(1\leqslant i\not=j\leqslant n)$ are Finite subsets of $F$ with cardinality $m$, and $A_1,\ldots, A_n$ are Finite Nonempty subsets of $F$ with $|A_1|=\cdots=|A_n|=k$. Let $p(F)$ be the additive order of the identity of $F$. It is proved that $|C|\geqslant \min\{p(F),\ \ n(k-1)-mn(n-1)+1\}$ if $p(F)>mn$. This conclusion refines the result of Hou and Sun.
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On restricted sumsets over a field
Finite Fields and Their Applications, 2014Co-Authors: Lilu ZhaoAbstract:We consider restricted sumsets over field F. Let C = { a 1 + ? + a n : a 1 ? A 1 , ? , a n ? A n , a i - a j ? S i j if i ? j } , where S i j ( 1 ≤ i ? j ≤ n ) are Finite subsets of F with cardinality m, and A 1 , ? , A n are Finite Nonempty subsets of F with | A 1 | = ? = | A n | = k . Let p ( F ) be the additive order of the identity of F. It is proved that | C | ? min ? { p ( F ) , n ( k - 1 ) - m n ( n - 1 ) + 1 } if p ( F ) m n . This conclusion refines the result of Hou and Sun 11.
Steven L. Kleiman - One of the best experts on this subject based on the ideXlab platform.
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Toward Clemens' Conjecture in degrees between 10 and 24
arXiv: Algebraic Geometry, 1996Co-Authors: Trygve Johnsen, Steven L. KleimanAbstract:We introduce and study a likely condition that implies the following form of Clemens' conjecture in degrees $d$ between 10 and 24: given a general quintic threefold $F$ in complex $\IP^4$, the Hilbert scheme of rational, smooth and irreducible curves $C$ of degree $d$ on $F$ is Finite, Nonempty, and reduced; moreover, each $C$ is embedded in $F$ with balanced normal sheaf $\O(-1)\oplus\O(-1)$, and in $\IP^4$ with maximal rank.
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Rational curves of degree at most 9 on a general quintic threefold
Communications in Algebra, 1996Co-Authors: Trygve Johnsen, Steven L. KleimanAbstract:ABSTRACT. We prove the following form of the Clemens conjecture in low degree. Let d ≤ 9, and let F be a general quintic threefold in P 4. Then (1) the Hilbert scheme of rational, smooth and irreducible curves of degree d on F is Finite, Nonempty, and reduced; moreover, each curve is embedded in F with normal bundle (−1) ⊕ (−1), and in P 4 with maximal rank. (2) On F, there are no rational, singular, reduced and irreducible curves of degree d, except for the 17,601,000 six-nodal plane quintics (found by Vainsencher). (3) On F, there are no connected, reduced and reducible curves of degree d with rational components.
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Rational curves of degree at most 9 on a general quintic threefold
arXiv: Algebraic Geometry, 1995Co-Authors: Trygve Johnsen, Steven L. KleimanAbstract:We prove the following form of the Clemens conjecture in low degree. Let $d\le9$, and let $F$ be a general quintic threefold in $\IP^4$. Then (1)~the Hilbert scheme of rational, smooth and irreducible curves of degree $d$ on $F$ is Finite, Nonempty, and reduced; moreover, each curve is embedded in $F$ with normal bundle $\O(-1)\oplus\O(-1)$, and in $\IP^4$ with maximal rank. (2)~On $F$, there are no rational, singular, reduced and irreducible curves of degree $d$, except for the 17,601,000 six-nodal plane quintics (found by Vainsencher). (3)~On $F$, there are no connected, reduced and reducible curves of degree $d$ with rational components.
Alfred Geroldinger - One of the best experts on this subject based on the ideXlab platform.
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a realization theorem for sets of lengths in numerical monoids
Forum Mathematicum, 2018Co-Authors: Alfred Geroldinger, Wolfgang A SchmidAbstract:We show that for every Finite Nonempty set L of integers greater than or equal to 2 there are a numerical monoid H and a squarefree element a ∈ H whose set of lengths L(a) is equal to L.
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A realization theorem for sets of distances
Journal of Algebra, 2017Co-Authors: Alfred Geroldinger, Wolfgang A SchmidAbstract:Abstract Let H be an atomic monoid. The set of distances Δ ( H ) of H is the set of all d ∈ N with the following property: there are irreducible elements u 1 , … , u k , v 1 … , v k + d such that u 1 ⋅ … ⋅ u k = v 1 ⋅ … ⋅ v k + d but u 1 ⋅ … ⋅ u k cannot be written as a product of l irreducible elements for any l ∈ N with k l k + d . It is well-known (and easy to show) that, if Δ ( H ) is Nonempty, then min Δ ( H ) = gcd Δ ( H ) . In this paper we show conversely that for every Finite Nonempty set Δ ⊂ N with min Δ = gcd Δ there is a Finitely generated Krull monoid H such that Δ ( H ) = Δ .
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A realization theorem for sets of distances
arXiv: Commutative Algebra, 2016Co-Authors: Alfred Geroldinger, Wolfgang A SchmidAbstract:Let $H$ be an atomic monoid. The set of distances $\Delta (H)$ of $H$ is the set of all $d \in \mathbb{N}$ with the following property: there are irreducible elements $u\_1, \ldots, u\_k, v\_1 \ldots, v\_{k+d}$ such that $u\_1 \cdot \ldots \cdot u\_k=v\_1 \cdot \ldots \cdot v\_{k+d}$ but $u\_1 \cdot \ldots \cdot u\_k$ cannot be written as a product of $\ell$ irreducible elements for any $\ell \in \mathbb{N}$ with $k\lt \ell \lt k+d$. It is well-known (and easy to show) that, if $\Delta (H)$ is Nonempty, then $\min \Delta (H) = \gcd \Delta (H)$. In this paper we show conversely that for every Finite Nonempty set $\Delta \subset \mathbb{N}$ with $\min \Delta = \gcd \Delta$ there is a Finitely generated Krull monoid $H$ such that $\Delta (H)=\Delta$.
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Minimal relations and catenary degrees in Krull monoids
arXiv: Commutative Algebra, 2016Co-Authors: Yushuang Fan, Alfred GeroldingerAbstract:Let $H$ be a Krull monoid with class group $G$. Then $H$ is factorial if and only if $G$ is trivial. Sets of lengths and sets of catenary degrees are well studied invariants describing the arithmetic of $H$ in the non-factorial case. In this note we focus on the set $Ca (H)$ of catenary degrees of $H$ and on the set $\mathcal R (H)$ of distances in minimal relations. We show that every Finite Nonempty subset of $\mathbb N_{\ge 2}$ can be realized as the set of catenary degrees of a Krull monoid with Finite class group. This answers Problem 4.1 of {arXiv:1506.07587}. Suppose in addition that every class of $G$ contains a prime divisor. Then $Ca (H)\subset \mathcal R (H)$ and $\mathcal R (H)$ contains a long interval. Under a reasonable condition on the Davenport constant of $G$, $\mathcal R (H)$ coincides with this interval and the maximum equals the catenary degree of $H$.
Wolfgang A Schmid - One of the best experts on this subject based on the ideXlab platform.
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a realization theorem for sets of lengths in numerical monoids
Forum Mathematicum, 2018Co-Authors: Alfred Geroldinger, Wolfgang A SchmidAbstract:We show that for every Finite Nonempty set L of integers greater than or equal to 2 there are a numerical monoid H and a squarefree element a ∈ H whose set of lengths L(a) is equal to L.
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A realization theorem for sets of distances
Journal of Algebra, 2017Co-Authors: Alfred Geroldinger, Wolfgang A SchmidAbstract:Abstract Let H be an atomic monoid. The set of distances Δ ( H ) of H is the set of all d ∈ N with the following property: there are irreducible elements u 1 , … , u k , v 1 … , v k + d such that u 1 ⋅ … ⋅ u k = v 1 ⋅ … ⋅ v k + d but u 1 ⋅ … ⋅ u k cannot be written as a product of l irreducible elements for any l ∈ N with k l k + d . It is well-known (and easy to show) that, if Δ ( H ) is Nonempty, then min Δ ( H ) = gcd Δ ( H ) . In this paper we show conversely that for every Finite Nonempty set Δ ⊂ N with min Δ = gcd Δ there is a Finitely generated Krull monoid H such that Δ ( H ) = Δ .
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A realization theorem for sets of distances
arXiv: Commutative Algebra, 2016Co-Authors: Alfred Geroldinger, Wolfgang A SchmidAbstract:Let $H$ be an atomic monoid. The set of distances $\Delta (H)$ of $H$ is the set of all $d \in \mathbb{N}$ with the following property: there are irreducible elements $u\_1, \ldots, u\_k, v\_1 \ldots, v\_{k+d}$ such that $u\_1 \cdot \ldots \cdot u\_k=v\_1 \cdot \ldots \cdot v\_{k+d}$ but $u\_1 \cdot \ldots \cdot u\_k$ cannot be written as a product of $\ell$ irreducible elements for any $\ell \in \mathbb{N}$ with $k\lt \ell \lt k+d$. It is well-known (and easy to show) that, if $\Delta (H)$ is Nonempty, then $\min \Delta (H) = \gcd \Delta (H)$. In this paper we show conversely that for every Finite Nonempty set $\Delta \subset \mathbb{N}$ with $\min \Delta = \gcd \Delta$ there is a Finitely generated Krull monoid $H$ such that $\Delta (H)=\Delta$.
