Multi-dimensional versions of a theorem of Fine and Wilf and a formula of Sylvester
| Autor: | Robert Tijdeman, R. J. Simpson |
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| Rok vydání: | 2003 |
| Předmět: | |
| Zdroj: | Proceedings of the American Mathematical Society. 131:1661-1671 |
| ISSN: | 1088-6826 0002-9939 |
| DOI: | 10.1090/s0002-9939-03-06970-3 |
| Popis: | Let v 0 → , . . . , v k → {\vec {v_0},..., \vec {v_k}} be vectors in Z k \mathbf {Z}^k which generate Z k \mathbf {Z}^k . We show that a body V ⊂ Z k V \subset \mathbf {Z}^k with the vectors v 0 → , . . . , v k → {\vec {v_0},..., \vec {v_k}} as edge vectors is an almost minimal set with the property that every function f : V → R f: V \rightarrow \mathbf {R} with periods v 0 → , . . . , v k → {\vec {v_0},..., \vec {v_k}} is constant. For k = 1 k=1 the result reduces to the theorem of Fine and Wilf, which is a refinement of the famous Periodicity Lemma. Suppose 0 → \vec {0} is not a non-trivial linear combination of v 0 → , . . . , v k → {\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 v 0 → , . . . , v k → {\vec {v_0},..., \vec {v_k}} over Z ≥ 0 \mathbf {Z}_{\geq 0} , but infinitely many points on each of its hyperfaces are not. For k = 1 k=1 the result reduces to a formula of Sylvester corresponding to Frobenius’ Coin-changing Problem in the case of coins of two denominations. |
| Databáze: | OpenAIRE |
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