Multi-dimensional versions of a theorem of Fine and Wilf and a formula of Sylvester期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Multi-dimensional versions of a theorem of Fine and Wilf and a formula of Sylvester

Authors:	R J Simpson R Tijdeman

Institution:	Department of Mathematics and Statistics, Curtin University of Technology, P.O. Box U1987, Perth, Western Australia 6001, Australia ; Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands

Abstract:	Let ${\vec {v_0},..., \vec {v_k}}$ be vectors in $\mathbf{Z}^k$ which generate $\mathbf{Z}^k$ . We show that a body $V \subset \mathbf{Z}^k$ with the vectors ${\vec {v_0},..., \vec {v_k}}$ as edge vectors is an almost minimal set with the property that every function $f: V \rightarrow \mathbf{R}$ with periods ${\vec {v_0},..., \vec {v_k}}$ is constant. For the result reduces to the theorem of Fine and Wilf, which is a refinement of the famous Periodicity Lemma. Suppose $\vec{0}$ is not a non-trivial linear combination of ${\vec {v_0},..., \vec {v_k}}$ with non-negative coefficients. Then we describe the sector such that every interior integer point of the sector is a linear combination of ${\vec {v_0},..., \vec {v_k}}$ over $\mathbf{Z}_{\geq 0}$ , but infinitely many points on each of its hyperfaces are not. For the result reduces to a formula of Sylvester corresponding to Frobenius' Coin-changing Problem in the case of coins of two denominations.

Keywords:	Periodicity Frobenius lattice coin-changing

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