Factorizations of the same length in abelian monoids
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García Barroso, Evelia R.
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García-Marco, Ignacio
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Márquez-Corbella, Irene
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Universidad de La Laguna
info
ISSN: 0035-5038, 1827-3491
Any de publicació: 2023
Volum: 72
Pàgines: 679-707
Tipus: Article
Altres publicacions en: Ricerche di Matematica
Resum
Let S⊆Zm⊕T be a finitely generated and reduced monoid. In this paper we develop a general strategy to study the set of elements in S having at least two factorizations of the same length, namely the ideal LS. To this end, we work with a certain (lattice) ideal associated to the monoid S. Our study can be seen as a new approach generalizing [9], which only studies the case of numerical semigroups. When S is a numerical semigroup we give three main results: (1) we compute explicitly a set of generators of the ideal LS when S is minimally generated by an almost arithmetic sequence; (2) we provide an infinite family of numerical semigroups such that LS is a principal ideal; (3) we classify the computational problem of determining the largest integer not in LS as an NP-hard problem.
Referències bibliogràfiques
- Abbott, J., Bigatti, A.M., Robbiano, L.: CoCoA: a system for doing Computations in Commutative Algebra. Available at http://cocoa.dima.unige.it
- Assi, A., García-Sánchez, P.A.: Numerical semigroups and applications. RSME Springer Series. Springer International Publishing, New York (2016)
- Bayer, D., Mumford, D.: What can be computed in algebraic geometry? In: Computational algebraic geometry and commutative algebra (Cortona, 1991), Sympos. Math., XXXIV, pages 1–48. Cambridge Univ. Press, Cambridge, (1993)
- Bermejo, I., García-Llorente, E., García-Marco, I.: Algebraic invariants of projective monomial curves associated to generalized arithmetic sequences. J. Symbolic Comput. 81, 1–19 (2017)
- Bermejo, I., García-Marco, I.: Complete intersections in certain affine and projective monomial curves. Bull. Braz. Math. Soc. (N.S.) 45(4), 599–624 (2014)
- Bigatti, A.M., La Scala, R., Robbiano, L.: Computing toric ideals. J. Symbolic Comput. 27(4), 351–365 (1999)
- Blanco, V., García-Sánchez, P.A., Geroldinger, A.: Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and krull monoids. Illinois J. of Math. 55, 1385–1414 (2011)
- Bruns, W., Gubeladze, J., Trung, N.V.: Problems and algorithms for affine semigroups. Semigroup Forum 64(2), 180–212 (2002)
- Chapman, S.T., García-Sánchez, P.A., Llena, D., Marshall, J.: Elements in a numerical semigroup with factorizations of the same length. Can. Math. Bull. 54(1), 39–43 (2011)
- Chapman, S. T., Coykendall, J., Gotti, F. , Smith, W.W. : Length-factoriality in commutative monoids and integral domains. J. Algebra (2021). arXiv:2101.05441 (to appear)
- Charalambous, H., Thoma, A., Vladoiu, M.: Minimal generating sets of lattice ideals. Collect. Math. 68(3), 377–400 (2017)
- Cimpoeaş, M., Stamate, D.I.: On intersections of complete intersection ideals. J. Pure Appl. Algebra 220(11), 3702–3712 (2016)
- Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups, Volume II. Number v. 2 in Mathematical surveys and monographs. American Mathematical Society, (1967)
- Cox, D.A., Little, J.B., Schenck, H.K.: Toric varieties.Graduate studies in mathematics. AMS - American Mathematical Society, USA (2011)
- Decker, W., Greuel, G-M., Pfister, G., Schönemann, H.: Singular 4-1-2- A computer algebra system for polynomial computations. http://www.singular.uni-kl.de, (2020)
- Eisenbud, D., Goto, S.: Linear free resolutions and minimal multiplicity. J. Algebra 88(1), 89–133 (1984)
- Foroutan, A., Hassler, W.: Chains of factorizations and factorizations with successive lengths. Commun. Algebra 34(3), 939–972 (2006)
- García-García, J.I., Marín-Aragón, D., Moreno-Frías, M.A.: Factorizations of the same length in numerical semigroups. Int. J. Comput. Math. 96(12), 2511–2521 (2019)
- García-Sánchez, P.A., Herrera-Poyatos, A.: Isolated factorizations and their applications in simplicial e semigroups. J.Algebra Appl. 19, 2050082 (2019)
- García-Sánchez, P.A., Ojeda, I., Rosales, J.C.: Affine semigroups having a unique Betti element. J. Algebra Appl. 12(3), 125–177 (2012)
- García-Sánchez, P.A., O’Neill, C., Webb, G.: The computation of factorization invariants for affine semigroups. J. Algebra Appl. 18(1), 1950019 (2019)
- Geroldinger, A., Halter-Koch, F.: Non-unique factorizations algebraic, combinatorial and analytic theory number 278 in pure and applied mathematics. Chapman and Hall /CRC, Ohio (2006)
- Geroldinger, A., Reinhart, A.: The monotone catenary degree of monoids of ideals. Int. J. Algebra Comput. 29(03), 419–457 (2019)
- Geroldinger, A., Zhong, Q.: Factorization theory in commutative monoids. Semigroup Forum 100(1), 22–51 (2020)
- Gonzalez, D., Wright, C., Zomback, J.: Monotone catenary degree in numerical monoids, (2019)
- Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/
- Hassler, W.: Properties of factorizations with successive lengths in one-dimensional local domains. J. Commut. Algebra 1(2), 237–268 (2009)
- Hemmecke, R., Malkin, P.N.: Computing generating sets of lattice ideals and Markov bases of lattices. J. Symbolic Comput. 44(10), 1463–1476 (2009)
- L’vovsky, S.: On inflection points, monomial curves, and hypersurfaces containing projective curves. Math. Ann. 306(4), 719–735 (1996)
- Márquez-Campos, G., Ojeda, I., Tornero, J.M.: On the computation of the apéry set of numerical monoids and affine semigroups. Semigroup Forum 91(1), 139–158 (2015)
- Papadimitriou, C.H., Steiglitz, K.: Combinatorial optimization: algorithms and complexity. Prentice-Hall Inc, Englewood Cliffs (1982)
- Philipp, A.: A characterization of arithmetical invariants by the monoid of relations ii: the monotone catenary degree and applications to semigroup rings. Semigroup Forum 90(1), 220–250 (2015)
- Pisón-Casares, P.: The short resolution of a lattice ideal. Proc. Amer. Math. Soc. 131(4), 1081–1091 (2002)
- Ramírez-Alfonsín, J.L.: Complexity of the Frobenius problem. Combinatorica 16(1), 143–147 (1996)
- Ramírez-Alfonsín, J.L.: The Diophantine Frobenius Problem, volume 30 of Oxford Lecture Series in Mathematics and Its Applications. OUP Oxford, (2005)
- Sturmfels, B.: Grobner bases and convex polytopes. Memoirs of the American Mathematical Society. American Mathematical Society, USA (1996)
- Villarreal, R.H.: Monomial algebras. Monographs and research notes in mathematics, 2nd edn. CRC Press, Boca Raton (2015)
- Vu, T.: Periodicity of Betti numbers of monomial curves. J. Algebra 418, 66–90 (2014)