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Numerical Semigroups and Applications by Abdallah Assi, Pedro A. García-Sánchez

By Abdallah Assi, Pedro A. García-Sánchez

This paintings offers purposes of numerical semigroups in Algebraic Geometry, quantity conception, and Coding concept. history on numerical semigroups is gifted within the first chapters, which introduce easy notation and basic suggestions and irreducible numerical semigroups. the point of interest is specifically on loose semigroups, that are irreducible; semigroups linked to planar curves are of this sort. The authors additionally introduce semigroups linked to irreducible meromorphic sequence, and express how those are utilized in order to offer the houses of planar curves. Invariants of non-unique factorizations for numerical semigroups also are studied. those invariants are computationally available during this surroundings, and therefore this monograph can be utilized as an creation to Factorization idea. due to the fact factorizations and divisibility are strongly hooked up, the authors convey a few purposes to AG Codes within the ultimate part. The publication can be of worth for undergraduate scholars (especially these at a better point) and in addition for researchers wishing to target the country of artwork in numerical semigroups research.

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We want to improve Proposition 19 when f is irreducible. In particular, we will prove that we can choose m = n in that proposition. To this end we will make use of minimal polynomials and algebraic extensions. In this way we will have a decomposition of f in an extension field that can be handled easier. Lemma 12 Let m ∈ N∗ . The extension K((t m )) → K((t)) is an algebraic extension of degree m. Proof The field K((t)) is a K((t m ))-vector space with basis {1, t, . . , t m−1 }. The proof now follows from [54, Theorem 46].

For all k ∈ {1, . . , h − 1}, (G 1 , . . , G k ) is a set of pseudo-approximate roots of G k+1 . Proof Fix k ∈ {1, . . , h − 1} and let i ∈ {1, . . , k}. By Proposition 28, int(G k+1 , G i ) = 1 ri int( f, G i ) = . dk+1 dk+1 Furthermore, G i is irreducible by Proposition 27, and we are done by definition. Let d be a divisor of n, and let G ∈ K((x))[y] be a monic polynomial of degree in y. Then the G-adic expansion of f has the form f = G d + α1 G d−1 + · · · + αd , n d 44 3 Semigroup of an Irreducible Meromorphic Series with αk ∈ K((x))[y] and deg y αi (x, y) < dn for all i ∈ {1, .

In the literature, sometimes these are chosen to be the definition of symmetric and pseudo-symmetric numerical semigroups. Proposition 14 Let S be a numerical semigroup. (i) S is symmetric if and only if for all x ∈ Z \ S, we have F(S) − x ∈ S. (ii) S is pseudo-symmetric if and only if F(S) is even and for all x ∈ Z \ S, either . F(S) − x ∈ S or x = F(S) 2 Proof (i) Assume that S is symmetric. Then F(S) is odd, and thus H = {x ∈ Z \ S | F(S) − x ∈ / S} = {x ∈ Z \ S | F(S) − x ∈ / S, x = F(S)/2}.

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