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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Commutative algebra of generalised Frobenius numbers
AU - Smith, Ben
AU - Manjunath, Madhusudan
PY - 2019/2/4
Y1 - 2019/2/4
N2 - We study commutative algebra arising from generalised Frobenius numbers. The k-th (generalised) Frobenius number of relatively prime natural numbers (a1,⋯,an) is the largest natural number that cannot be written as a non-negative integral combination of (a1,⋯,an) in k distinct ways. Suppose that L is the lattice of integer points of (a1,⋯,an)⊥. Taking our cue from the concept of lattice modules due to Bayer and Sturmfels, we define generalised lattice modules M(k)L whose Castelnuovo–Mumford regularity captures the k-th Frobenius number of (a1,⋯,an). We study the sequence {M(k)L} of generalised lattice modules providing an explicit characterisation of their minimal generators. We show that there are only finitely many isomorphism classes of generalised lattice modules. As a consequence of our commutative algebraic approach, we show that the sequence of generalised Frobenius numbers forms a finite difference progression i.e. a sequence whose set of successive differences is finite. We also construct an algorithm to compute the k-th Frobenius number.
AB - We study commutative algebra arising from generalised Frobenius numbers. The k-th (generalised) Frobenius number of relatively prime natural numbers (a1,⋯,an) is the largest natural number that cannot be written as a non-negative integral combination of (a1,⋯,an) in k distinct ways. Suppose that L is the lattice of integer points of (a1,⋯,an)⊥. Taking our cue from the concept of lattice modules due to Bayer and Sturmfels, we define generalised lattice modules M(k)L whose Castelnuovo–Mumford regularity captures the k-th Frobenius number of (a1,⋯,an). We study the sequence {M(k)L} of generalised lattice modules providing an explicit characterisation of their minimal generators. We show that there are only finitely many isomorphism classes of generalised lattice modules. As a consequence of our commutative algebraic approach, we show that the sequence of generalised Frobenius numbers forms a finite difference progression i.e. a sequence whose set of successive differences is finite. We also construct an algorithm to compute the k-th Frobenius number.
U2 - 10.5802/alco.31
DO - 10.5802/alco.31
M3 - Journal article
VL - 2
SP - 149
EP - 171
JO - Algebraic Combinatorics
JF - Algebraic Combinatorics
IS - 1
ER -