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SUMMARY:David Grynkiewicz (Memphis University)
DTSTART:20260713T160000Z
DTEND:20260713T162500Z
DTSTAMP:20260710T111436Z
UID:CANT2026/7
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2026/7/"
 >On factorizations of zero-sum sequences over  abelian torsion groups</a>\
 nby David Grynkiewicz (Memphis University) as part of Combinatorial and ad
 ditive number theory seminar (CANT 2026)\n\nLecture held in Science Center
  in the CUNY Graduate Center (4th floor).\n\nAbstract\nLet $G$ be an addit
 ive abelian torsion group and let $G_0\\subseteq G$ be a subset. A zero-su
 m sequence over $G_0$ is an unordered string of terms from $G_0$ (repetiti
 on of terms allowed) such that the sum of terms is $0$. In the last few de
 cades\, the connection between factorizations of zero-sum sequences and fa
 ctorization of elements in rings of integers has been made more precise an
 d extended into much more general algebraic settings. The extent to which 
 factorization are wild or well-behaved is often measured by the finiteness
  and size of various arithmetic factorization invariants. Some of the most
  common include the catenary degree $\\mathsf c(G_0)$\, the set of success
 ive distances $\\Delta(G_0)$\, and the elastacities $\\rho_k(G_0)$. We beg
 in by introducing what these invariants are in purely combinatorial terms 
 and explain how they measure constraint of factorization in algebraic sett
 ings. \n\nIn the past\, there has been much focus on finite groups\, and m
 ore recently\, on subsets of finitely generated groups. However\, very lit
 tle was known in the case of non-finitely generated abelian groups. In par
 t\, this is because common invariants used to study factorization\, such a
 s the Davenport Constant\, are no longer guaranteed to be finite. In order
  to better understand factorization in the setting of infinite abelian tor
 sion groups\, we introduce a new technique measuring the size of a sequenc
 e not by the number of its terms but rather by its cross number\, $\\sum_{
 i=1}^{\\ell} \\frac{1}{\\text{\\rm ord} (g_i)}$\, where the $g_i\\in G_0\\
 subseteq G$ are the terms in the sequence. The use of cross numbers allows
  us to define three constants\, $\\mathsf K(G_0)$\, $\\mathsf k(G_0)$ and 
 $\\mathsf K_{\\mathsf{inf}}(G_0)$\, defined as the supremum of all cross n
 umbers of minimal (by inclusion) zero-sum sequences\, the supremum of all 
 cross numbers of zero-sum free sequences (sequences having no zero-sum sub
 sequence)\, and the infimum of all cross numbers of nontrivial zero-sum se
 quences. The first two of these constants have appeared in the literature 
 before\, but the third is newly introduced here. \n\nIn the first part of 
 this two part talk\, it was shown that factorization of zero-sum sequences
  can be very ill-behaved when $\\mathsf K_{\\mathsf{inf}}(G_0)=0$. In this
  second part\, we consider what happens when $\\mathsf K_{\\mathsf{inf}}(G
 _0)>0$\, specifically in the setting of infinite abelian torsion groups wi
 th finite total rank. In this setting\, the first two cross number constan
 ts $\\mathsf K(G_0)$ and $\\mathsf k(G_0)$ are always finite. Assuming $\\
 delta:=\\mathsf K_{\\mathsf{inf}}(G_0)>0$\, we then obtain a general upper
  bound for the catenary degree $$\\mathsf c(G_0)\\leq \\max\\{2\\delta^{-1
 }\\mathsf k(G_0)+1\, \\quad 2\\delta^{-1}\\mathsf K(G_0)\\}.$$ In particul
 ar\, this implies that both the set of successive distances $\\Delta(G_0)$
  and catenary degree are always finite under these circumstances\, with ex
 plicit concrete upper bounds. Moreover\, our upper bound on the catenary d
 egree is tight\, meaning there are infinite families of subsets $G_0\\subs
 eteq G$ for which equality holds above. In addition\, for the special case
  of quasi-cyclic groups\, we are able to partially characterize what subse
 ts $G_0$ with $\\mathsf K_{\\mathsf{inf}}(G_0)>0$ look like and use this t
 o give a lower bound for the elasticities $\\rho_k(G_0)$. Combined with th
 e upper bound on the catenary degree\, this yields a structural descriptio
 n of the possible refactorization lengths of a product of $k$ irreducibles
 . This is joint work with Alfred Geroldinger and Guoqing Wang.\n
LOCATION:https://researchseminars.org/talk/CANT2026/7/
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