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.

**Read or Download Numerical Semigroups and Applications PDF**

**Best machine theory books**

**Digital and Discrete Geometry: Theory and Algorithms**

This publication presents complete assurance of the trendy equipment for geometric difficulties within the computing sciences. It additionally covers concurrent subject matters in facts sciences together with geometric processing, manifold studying, Google seek, cloud information, and R-tree for instant networks and BigData. the writer investigates electronic geometry and its similar optimistic equipment in discrete geometry, supplying designated tools and algorithms.

This ebook constitutes the refereed court cases of the twelfth overseas convention on man made Intelligence and Symbolic Computation, AISC 2014, held in Seville, Spain, in December 2014. The 15 complete papers offered including 2 invited papers have been conscientiously reviewed and chosen from 22 submissions.

This ebook constitutes the refereed lawsuits of the 3rd overseas convention on Statistical Language and Speech Processing, SLSP 2015, held in Budapest, Hungary, in November 2015. The 26 complete papers offered including invited talks have been rigorously reviewed and chosen from seventy one submissions.

- Ensemble Methods: Foundations and Algorithms (Chapman & Hall/CRC Data Mining and Knowledge Discovery Serie)
- Higher-Order Computability (Theory and Applications of Computability)
- The Universe as Automaton: From Simplicity and Symmetry to Complexity (SpringerBriefs in Complexity)
- A Hybrid Deliberative Layer for Robotic Agents: Fusing DL Reasoning with HTN Planning in Autonomous Robots (Lecture Notes in Computer Science)

**Additional info for Numerical Semigroups and Applications**

**Example text**

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}.