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SUMMARY:Igor Shparlinski (UNSW\, Australia)
DTSTART:20220421T213000Z
DTEND:20220421T223000Z
DTSTAMP:20260423T024022Z
UID:JNTS/44
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/JNTS/44/">Ch
 aracteristic Polynomials and Multiplicative Dependence of Integer Matrices
 </a>\nby Igor Shparlinski (UNSW\, Australia) as part of Columbia CUNY NYU 
 number theory seminar\n\n\nAbstract\nWe consider the set  $\\mathcal{M}_n(
 \\mathbb{Z}\; H))$ of $n\\times n$-matrices with \ninteger elements of siz
 e at most $H$ and obtain  upper and lower bounds on the number of $s$-tupl
 es\nof matrices  from $\\mathcal{M}_n(\\mathbb{Z}\; H)$\, satisfying  vari
 ous multiplicative relations\, including \n  multiplicative  dependence\, 
 commutativity and \n bounded generation of a subgroup of $\\text{\\rm GL}_
 n(\\mathbb{Q})$.  These problems generalise those studied \nin the scalar 
 case $n=1$ by F. Pappalardi\, M. Sha\, I. E. Shparlinski and C. L. Stewart
  (2018) with an \nobvious distinction due to the non-commutativity of matr
 ices. \nAs a part of our method\, we obtain  a new upper bound on the numb
 er of matrices  from $\\mathcal{M}_n(\\mathbb{Z}\; H)$\nwith a given chara
 cteristic polynomial $f \\in\\mathbb{Z}[X]$\,  which is uniform with respe
 ct to $f$. This complements \nthe asymptotic formula of A. Eskin\, S. Moze
 s and  N. Shah (1996) in which $f$ has to be fixed and irreducible. \n\nJo
 int work with Alina Ostafe.\n
LOCATION:https://researchseminars.org/talk/JNTS/44/
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