Author, as appears in the article.: Bras-Amorós, M
Department: Enginyeria Informàtica i Matemàtiques
URV's Author/s: Bras Amoros, Maria
Keywords: Conjecture
Abstract: We present a revisit of the seeds algorithm to explore the semigroup tree. First, an equivalent definition of seed is presented, which seems easier to manage. Second, we determine the seeds of semigroups with at most three left elements. And third, we find the great-grandchildren of any numerical semigroup in terms of its seeds. The the right-generators descendant (RGD) algorithm is the fastest known algorithm at the moment. But if one compares the originary seeds algorithm with the RGD algorithm, one observes that the seeds algorithm uses more elaborated mathematical tools while the RGD algorithm uses data structures that are better adapted to the final C implementations. For genera up to around one half of the maximum size of native integers, the newly defined seeds algorithm performs significantly better than the RGD algorithm. For future compilators allowing larger native sized integers this may constitute a powerful tool to explore the semigroup tree up to genera never explored before. The new seeds algorithm uses bitwise integer operations, the knowledge of the seeds of semigroups with at most three left elements and of the great-grandchildren of any numerical semigroup, apart from techniques such as parallelization and depth first search as wisely introduced in this context by Fromentin and Hivert [Math. Comp. 85 (2016) pp. 2553-2568]. The algorithm has been used to prove that there are no Eliahou semigroups of genus 66, hence proving the Wilf conjecture for genus up to 66. We also found three Eliahou semigroups of genus 67. One of these semigroups is neither of Eliahou-Fromentin type, nor of Delgado's type. However, it is a member of a new family suggested by Shalom Eliahou.
Thematic Areas: Mathematics, applied Matemática / probabilidade e estatística Interdisciplinar Computational mathematics Ciência da computação Applied mathematics Algebra and number theory
licence for use: https://creativecommons.org/licenses/by/3.0/es/
Author's mail: maria.bras@urv.cat
Author identifier: 0000-0002-3481-004X
Record's date: 2024-08-03
Papper version: info:eu-repo/semantics/acceptedVersion
Link to the original source: https://www.ams.org/journals/mcom/2024-93-345/S0025-5718-2023-03881-9/
Licence document URL: https://repositori.urv.cat/ca/proteccio-de-dades/
Papper original source: Mathematics Of Computation. 93 (345): 411-441
APA: Bras-Amorós, M (2024). ON THE SEEDS AND THE GREAT-GRANDCHILDREN OF A NUMERICAL SEMIGROUP. Mathematics Of Computation, 93(345), 411-441. DOI: 10.1090/mcom/3881
Article's DOI: 10.1090/mcom/3881
Entity: Universitat Rovira i Virgili
Journal publication year: 2024
Publication Type: Journal Publications