Frobenius number of a linear Diophantine equation
Résumé
We denote by N₀ the set of nonnegative integers. Let d≥1 and A={a₁,...,a_{d}} a set of positive integers. For every n∈N₀, we write s(n) for the number of solutions (x₁,...,x_{d})∈N₀^{d} of the equation a₁x₁+⋯+a_{d}x_{d}=n. We set g(A)=sup{n∣s(n)=0}∪{-1} the Frobenius number of A. Let S(A) be the subsemigroup of (N₀,+) generated by A. We set S′(A)=N₀\S(A), N′(A)= CardS′(A) and N(A)= Card S(A)∩{0,1,..,g(A)}. Let p be a multiple of lcm(A) and F_{p}(t)=∏_{i=1}^{d}∑_{j=0}^{(p/(a_{i}))-1}t^{ja_{i}}. We give an upper bound for g(A) and reduction formulas for g(A),N′(A) and N(A). Characterizations of these invariants as well as numerical symmetric and pseudo-symmetric semigroups in terms of F_{p}(t), are also obtained.
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