ABSTRACT
We denote by ℕ0 the set of nonnegative integers. Let d ≥ 1 and A = {a 1, …, ad } a set of positive integers. For every n ∈ ℕ0, we write s(n) for the number of solutions https://www.w3.org/1998/Math/MathML"> ( x 1 , … , x d ) ∈ ℕ 0 d https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq54.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> of the equation a 1 x 1 +⋯+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 https://www.w3.org/1998/Math/MathML"> F p ( t ) = ∏ i = 1 d ∑ J = 0 P a i − 1 t j a i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq55.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . 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.
