TAILIEUCHUNG - Báo cáo toán học: "Factoring in embedding dimension three numerical semigroups"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài:Factoring in embedding dimension three numerical semigroups. | Factoring in embedding dimension three numerical semigroups F. Aguiló-Gost P. A. Garcia-Sanchez Dept. Matematica Aplicada IV Depto. de Algebra U. Politectina de Catalunya Barcelona Spain U. de Granada E-18071 Granada Spain matfag@ pedro@ Submitted Jan 5 2010 Accepted Sep 27 2010 Published Oct 15 2010 Mathematics Subject Classifications 05C90 11D07 11D45 11P21 Abstract Let us consider a 3-numerical semigroup S a b N . Given m E S the triple x y z E N3 is a factorization of m in S if xa yb zN m. This work is focused on finding the full set of factorizations of any m E S and as an application we compute the catenary degree of S. To this end we relate a 2D tessellation to S and we use it as a main tool. 1 Introduction Let us denote the set of non negative integers by N. A 3-numerical semigroup is the set S a b N xa yb zN x y z E N with 1 a b N and gcd a b N 1 such that the set a b N is the minimal set of generators of S. The set S N S has finite cardinality with Frobenius number f S max S. The Apery set of S with respect to m is the set Ap m S s E S s m E S . This set acts like a boundary between elements that can be factored in S and those that can not inside each equivalence class modulo m the reader can find in 19 an introduction to numerical semigroups . A factorization of m E S is a triple x y z E N3 such that xa yb zN m. Let us denote F m S x y z E N3 xa yb zN m and d m S F m S also known as the denumerant of m in S. See 17 for an exhaustive view of related results. Example 1 For S 3 5 7 we have f S 4 and Ap 7 S 0 3 5 6 8 9 11 . Consider m 15 we have d m S 3 and F m S 5 0 0 0 3 0 1 1 1 . -Work supported by MCYT ref. MTM2008-06620-C03-01 MTM and the Catalan Research Council under the projects DURSI 2005SGR00256 and 2009SGR1387. tWork supported by MCYT ref. MTM2007-62346 and MTM2010-15595 THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R138 1 Let x y z G F m S . The length of x y z is x y z x y z. For x y z x y z G N3 write crcd x 7 z xk 7 minix. x

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