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  • Author or Editor: Gábor Hegedűs x
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Let n be an arbitrary integer, let p be a prime factor of n . Denote by ω 1 the p th primitive unity root,

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\omega _1 : = e^{\tfrac{{2\pi i}} {p}}$$ \end{document}
.Define ω iω 1 i for 0 ≦ ip − 1 and B ≔ {1, ω 1 , …, ω p −1 } n ⊆ ℂ n .Denote by K ( n; p ) the minimum k for which there exist vectors ν 1 , …, ν kB such that for any vector wB , there is an i , 1 ≦ ik , such that ν i · w = 0, where ν · w is the usual scalar product of ν and w .Gröbner basis methods and linear algebra proof gives the lower bound K ( n; p ) ≧ n ( p − 1).Galvin posed the following problem: Let m = m ( n ) denote the minimal integer such that there exists subsets A 1 , …, A m of {1, …, 4 n } with | A i | = 2 n for each 1 ≦ in , such that for any subset B ⊆ [4 n ] with 2 n elements there is at least one i , 1 ≦ im , with A iB having n elements. We obtain here the result m ( p ) ≧ p in the case of p > 3 primes.

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Let F be a field, and α0,...,αk-1 be k distinct elements of F. Let λ =(λ1,...,λk) be a partition of n and V λ be the set of all vectors v=(v 1,...,v n)∈ F n such that  |{j ∈ [n] : v ji}|=λi+1  for 0≦ i ≦\ k-1. We describe the lexicographic standard monomials of the ideal of polynomials from  F[x 1,...,x n] which vanish on the set V λ. In the proof we give a new description of the orthogonal complement (S λ) (with respect to the James scalar product) of the Specht module S λ. As applications, a basis of (S λ) is exhibited, and we obtain a combinatorial description of the Hilbert function of V λ..  Our approach gives also the deglex standard monomials of V λ, and hence provides a new proof of a result of A. M. Garsia and C. Procesi [10].

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R. P. Stanley proved the Upper Bound Conjecture in 1975. We imitate his proof for the Ehrhart rings.

We give some upper bounds for the volume of integrally closed lattice polytopes. We derive some inequalities for the δ-vector of integrally closed lattice polytopes. Finally we apply our results for reflexive integrally closed and order polytopes.

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