By Abdallah Assi, Pedro A. García-Sánchez

This paintings offers functions of numerical semigroups in Algebraic Geometry, quantity conception, and Coding thought. history on numerical semigroups is gifted within the first chapters, which introduce uncomplicated notation and basic innovations and irreducible numerical semigroups. the point of interest is specifically on unfastened semigroups, that are irreducible; semigroups linked to planar curves are of this type. The authors additionally introduce semigroups linked to irreducible meromorphic sequence, and exhibit how those are utilized in order to give the homes of planar curves. Invariants of non-unique factorizations for numerical semigroups also are studied. those invariants are computationally obtainable during this environment, and hence this monograph can be utilized as an creation to Factorization idea. due to the fact that factorizations and divisibility are strongly attached, the authors exhibit a few purposes to AG Codes within the ultimate part. The publication could be of worth for undergraduate scholars (especially these at the next point) and likewise for researchers wishing to target the nation of paintings in numerical semigroups research.

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**Example text**

Proof Let k ∈ {1, . . , h} and let G k be a dkth pseudo-approximate root of f . Uniqueness in Lemma 19 allows us to repeat the first part of the proof starting with G k instead of y n/dk . By Proposition 28, int( f, G k ) = int( f, T(G k )). But App( f, dk ) can be obtained by applying the operation T finitely many times to G k . Hence the result is a consequence of Proposition 29 and Corollary 13. Corollary 14 For all k ∈ {1, . . , h}, App( f, dk ) is irreducible. In particular, App( f, dk ) is a dkth pseudo-approximate root of f .

Proposition 22 Under the standing hypothesis. (i) The sequence of Newton–Puiseux exponents of G k is given by m1 , . . , mdk−1 dk k (ii) The r -sequence and d-sequence of G k are given, respectively, by and d0 , . . , ddk−1 ,1 dk k . r0 , . . , rk−1 dk dk . Proof (i) This follows from the expression of Y (t) and the definition of Newton– Puiseux exponents, using the fact that gcd dnk , . . , mdk−1 is one. k (ii) Let R, D, E be the characteristic sequences associated with G k . We have D1 = R0 = deg y G k = dnk = dr0k = ddk1 , R1 = mdk1 = dr1k and D2 = gcd( dr0k , dr1k ) = dd2k .

We proceed as in Lemma 19 to compute App( f ; 2). We start with G = y 6/2 = y 3 . The G-adic expansion of f is G 2 + (−2x 2 )G + (x 4 − x 5 y). Hence α1 = −2x 2 = 0. So, we need to compute G 1 = T(G) = y 3 − x 2 . The G 1 -adic expansion of f is f = G 21 + (−x 5 y), and now α11 = 0. This means that g = G 1 = y 3 − x 2 = App( f ; 2). The picture below is in R2 : the dashed line corresponds to g. 2 Characteristic Sequences 47 y x Thus we have an algorithmic method to calculate approximate roots of f .