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SUMMARY:Shruti S Hegde (Ramakrishna Mission Vivekananda Educational and Re
 search Institute\, India)
DTSTART:20220525T130000Z
DTEND:20220525T132500Z
DTSTAMP:20260423T011436Z
UID:CANT2022/15
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2022/15/
 ">Weighted zero-sum constants and inverse results</a>\nby Shruti S Hegde (
 Ramakrishna Mission Vivekananda Educational and Research Institute\, India
 ) as part of Combinatorial and additive number theory (CANT 2022)\n\n\nAbs
 tract\nA weighted generalization of classical zero-sum constants\nwas intr
 oduced by Adhikari et al. in 2006 and has been an active area of research 
 since then. In the last fifteen years\, weighted zero-sum constants for $\
 \mathbb {Z}_n$ with several interesting weight sets have been found.\nIn t
 his talk\, we take up the problem of determining the exact values and prov
 iding bounds of weighted Davenport constant of $\\mathbb {Z}_n$  \nwith so
 me new weight sets.\n\nNext\, we consider a weighted generalization of the
  Erd\\H{o}s-Ginzburg-Ziv constant. \nLet $G$ be a  finite abelian group wi
 th $\\exp(G)=n$. For a positive integer $k$ and a non-empty subset $A$ of 
 $[1\, n-1]$\,\nthe arithmetical invariant $\\mathsf s_{kn\,A}(G)$  is defi
 ned to be the  least positive integer $t$ such that\nany sequence of $t$ e
 lements in $G$ has an $A$- weighted zero-sum subsequence of  length $kn$.\
 nWe give the exact value of $\\mathsf s_{kq\,A}(G)$\, for integers $k\\geq
  2$ and $A=\\{1\,2\\}$\,\nwhen $G$ is an abelian $p$-group with $rank(G)\\
 leq 4$\, $p$ is an odd prime  and $exp(G)=q$.\nOur method consists of a mo
 dification of a polynomial method \nof R\\'onyai.\n\nLastly\, we consider 
 the questions regarding inverse problems for the weighted zero-sum constan
 ts of $\\mathbb {Z}_n$. An inverse problem is a problem of characterizing 
 all the weighted {\\it zero-sum free sequences} over $\\mathbb {Z}_n$ of s
 pecific lengths for the particular weight sets under consideration.\n\nThi
 s work was joint with Sukumar Das Adhikari and partly with Md Ibrahim Moll
 a and Subha Sarkar.\n
LOCATION:https://researchseminars.org/talk/CANT2022/15/
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