2011
DOI: 10.2140/involve.2011.4.187
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The multidimensional Frobenius problem

Abstract: We provide a variety of results concerning the problem of determining maximal vectors g such that the Diophantine system M x = g has no solution: conditions for the existence of g, conditions for the uniqueness of g, bounds on g, determining g explicitly in several important special cases, constructions for g, and a reduction for M.

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Cited by 4 publications
(6 citation statements)
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References 14 publications
(10 reference statements)
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“…As mentioned in the introduction, it was also shown in [17] that the C +u−v is the unique maximal cone with the property that all points in the interior admit a non-negative integral representation, but infinitely many on the boundary do not have such a representation. This seems to be a very particular property of the case m = n − 1 as it was already pointed out in [4]. We want to conclude the paper with another example showing that, in general, we have more than one maximal cone.…”
Section: Proof Of Theorem 11supporting
confidence: 56%
See 1 more Smart Citation
“…As mentioned in the introduction, it was also shown in [17] that the C +u−v is the unique maximal cone with the property that all points in the interior admit a non-negative integral representation, but infinitely many on the boundary do not have such a representation. This seems to be a very particular property of the case m = n − 1 as it was already pointed out in [4]. We want to conclude the paper with another example showing that, in general, we have more than one maximal cone.…”
Section: Proof Of Theorem 11supporting
confidence: 56%
“…inclusion) translated cone h + C such that all integral points in its interior belong to F 1 (A), but there are (infinitely many) integral points on its boundary which are not in F 1 (A). In general, i.e., for 1 < m < n − 1, such a cone is not uniquely determined, see, for instance, [4] and Section 5.…”
Section: Introduction and Statement Of Resultsmentioning
confidence: 99%
“…, t k ∈ Q ≥0 such that r k+1 = t 1 r 1 + · · · + t k r k . From this it easily follows (2) implies (1). Assume that there exist q 1 , .…”
Section: Gluings and Conesmentioning
confidence: 86%
“…First notice that if we pick t greater than the maximal expression in the upper bound of (4), then (19) implies G Λa (t) > s. In addition, (28) implies that for s ≤ τ N −1 + 1, g s (a) satisfies (5). As for lower bounds on g s (a), if we pick…”
Section: Bounds On G S (A)mentioning
confidence: 99%
“…It should also be mentioned that other generalizations of the Frobenius number of different nature have also been considered by a variety of authors. In particular, see Chapter 6 of [1], as well as more recent works [4], [5], and [23], among others, for further information and references.…”
Section: Introductionmentioning
confidence: 99%