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SUMMARY:Alina Ostafe (The University of New South Wales)
DTSTART:20220610T103000Z
DTEND:20220610T113000Z
DTSTAMP:20260423T022142Z
UID:ntsea/16
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/ntsea/16/">I
 nteger matrices with a given characteristic polynomial and multiplicative 
 dependence of matrices</a>\nby Alina Ostafe (The University of New South W
 ales) as part of Number theory by the sea\n\n\nAbstract\nWe consider the s
 et $\\mathcal{M}_n(\\mathbb{Z}\; H))$ of $n\\times n$-matrices with\ninteg
 er elements of size at most $H$ and obtain upper and lower bounds on the n
 umber\nof $s$-tuples\nof matrices from $\\mathcal{M}_n(\\mathbb{Z}\; H)$\,
  satisfying various multiplicative\nrelations\, including\nmultiplicative 
 dependence\, commutativity and\nbounded generation of a subgroup of $\\mat
 hrm{GL}_n(\\mathbb{Q})$. These problems\ngeneralise those studied\nin the 
 scalar case $n=1$ by F. Pappalardi\, M. Sha\, I. E. Shparlinski and C. L.\
 nStewart (2018) with an\nobvious distinction due to the non-commutativity 
 of matrices.\nAs a part of our method\, we obtain a new upper bound on the
  number of matrices from\n$\\mathcal{M}_n(\\mathbb{Z}\; H)$\nwith a given 
 characteristic polynomial $f \\in\\mathbb{Z}[X]$\, which is uniform with\n
 respect to $f$. This complements\nthe asymptotic formula of A. Eskin\, S. 
 Mozes and N. Shah (1996) in which $f$ has to\nbe fixed and irreducible.\n\
 nJoint work with Igor Shparlinski.\n
LOCATION:https://researchseminars.org/talk/ntsea/16/
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