The Diophantine Frobenius problem

259 Pages·2005·1.116 MB·English

by  Jorge L. Ramírez Alfonsín

Oxford Lecture Series in
Mathematics and its Applications 30
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30. Jorge Ramirez Alfonsin: The Diophantine Frobenius Problem
The Diophantine
Frobenius Problem
J.L. Ram´ırez Alfons´ın
Equipe Combinatoire
Universit´e Pierre et Marie Curie, Paris 6
4 Place Jussieu, 75252 Paris Cedex 05
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Contents
Preface xi
Acknowledgements xv
1 Algorithmic aspects 1
1.1 Algorithms for computing g(a ,a ,a ) 3
1 2 3
1.1.1 Rødseth’s Algorithm 3
1.1.2 Davison’s Algorithm 4
1.1.3 Killingbergtrø’s method 6
1.2 General algorithms 8
1.2.1 Scarf and Shallcross’ method 8
1.2.2 Heap and Lynn method 11
1.2.3 Greenberg’s Algorithm 18
1.2.4 Nijenhuis’ Algorithm 19
1.2.5 Wilf’s Algorithm 19
1.2.6 Kannan’s method 21
1.3 Computational complexity of FP 24
1.4 Supplementary notes 28
2 The Frobenius number for small n 31
2.1 Computing g(p,q) is easy 31
2.2 A Formula for g(a ,a ,a ) 35
1 2 3
2.3 Results when n = 3 36
2.3.1 Hofmeister’s formula and its generalization 39
2.3.2 More special cases 40
2.3.3 Johnson integers 42
2.4 g(a ,a ,a ,a ) 42
1 2 3 4
2.5 Supplementary notes 43
3 The general problem 45
3.1 Formulas and upper bounds 45
3.2 Bounds in terms of the lcm(a ,...,a ) 57
1 n
3.3 Arithmetic and related sequences 59
3.4 Regular bases 61
3.5 Extending basis 62
viii Contents
3.6 Lower bounds 63
3.7 Supplementary notes 68
4 Sylvester denumerant 71
4.1 From partitions to denumerants 71
4.2 Formulas and bounds for d(m;a ,...,a ) 73
1 n
4.3 Computing denumerants 77
4.3.1 Partial fractions 77
4.3.2 Bell’s method 78
4.4 d(m;p,q) 80
4.5 d(m;a ,a ,a ) and d(m;a ,a ,a ,a ) 81
1 2 3 1 2 3 4
4.6 Hilbert series 86
4.7 A proof of a formula for g(a ,a ,a ) 89
1 2 3
4.8 Ehrhart polynomial 91
4.9 Variations of the denumerant 95
4.9.1 d(cid:2)(m;a ,...,a ) 96
1 n
4.9.2 d(cid:2)(cid:2)(m;a ,...,a ) 98
1 n
4.10 Supplemetary notes 100
5 Integers without representation 103
5.1 Sylvester’s classical result 103
5.2 Nijenhuis’ and Wilf’s results 105
5.3 Formulas for N(a ,...,a ) 108
1 n
5.4 Arithmetic sequences 110
5.5 The sum of integers in N(p,q) 111
5.6 Related games 113
5.6.1 Sylver Coinage 113
5.6.2 The jugs problem 114
5.7 Supplemetary notes 117
6 Generalizations and related problems 119
6.1 Special functions 119
6.2 The modular generalization 124
6.3 The postage stamp problem 127
6.4 (a ,...,a )-trees 128
1 n
6.5 Vector generalization of FP 130
6.6 Supplementary notes 133
7 Numerical semigroups 135
7.1 Gaps and non-gaps 135
7.1.1 Telescopic semigroups 139
7.1.2 Hyperelliptic semigroups 140
Contents ix
7.2 Symmetric semigroups 141
7.2.1 Intersection of semigroups 148
7.2.2 Ap´ery sets 149
7.3 Related concepts 150
7.3.1 Type sequences 150
7.3.2 Complete intersection 152
7.3.3 The Mo¨bius function 153
7.4 Supplementary notes 154
8 Applications of the Frobenius number 159
8.1 Complexity analysis of the Shell-sort method 159
8.2 Petri Nets 161
8.2.1 P/T systems 161
8.2.2 Weighted circuits systems 163
8.3 Partition of a vector space 165
8.4 Monomial curves 168
8.5 Algebraic geometric codes 171
8.6 Tilings 174
8.7 Applications of denumerants 175
8.7.1 Balls and cells 175
8.7.2 Conjugate power equations 177
8.7.3 Invariant cubature formulas 179
8.8 Other applications 179
8.8.1 Generating random vectors 179
8.8.2 Non-hamiltonian graphs 181
8.9 Supplementary notes 183
Appendix A
Problems and conjectures 185
A.1 Algorithmic questions 185
A.2 g(a ,...,a ) 186
1 n
A.3 Denumerant 187
A.4 N(a ,...,a ) 187
1 n
A.5 Gaps 188
A.6 Miscellaneous 189
A.6.1 Erdo˝s’ Problems 191
Appendix B 193
B.1 Computational complexity aspects 193
B.2 Graph theory aspects 194
B.3 Modules, resolutions and Hilbert series 194
B.4 Shell-sort method 197
x Contents
B.5 Bernoulli numbers 198
B.6 Irreducible and primitive matrices 200
B.6.1 Upper bounds of index of primitivity 202
B.6.2 Computation of index of primitivity 203
References 205
Index 241