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SUMMARY:Leonid Fel (Technion -- Israel Institute of Technology\, Israel)
DTSTART:20220525T140000Z
DTEND:20220525T142500Z
DTSTAMP:20260423T011437Z
UID:CANT2022/18
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2022/18/
 ">Commutative monoid of self-dual symmetric polynomials</a>\nby Leonid Fel
  (Technion -- Israel Institute of Technology\, Israel) as part of Combinat
 orial and additive number theory (CANT 2022)\n\n\nAbstract\nWe consider a 
 set ${\\mathfrak R}{\\mathfrak S}\\left(\\lambda\,S_n\\right)$ of self-\na
 nd skew-reciprocal polynomials in $\\lambda$\, of degree $mn$\, where $m\\
 in{\n\\mathbb Z}_{\\geq}$\, $n\\in{\\mathbb Z}_>$\, based on polynomial in
 variants $I_{n\,\nr}({\\bf x}^n)$ of symmetric group $S_n$\, acting on the
  Euclidean space ${\\mathbb\nE}^n$ over the field of real numbers ${\\math
 bb R}$\, where ${\\bf x^n}=\\{x_1\,\n\\ldots\,x_n\\}\\in{\\mathbb E}^n$. W
 e prove that ${\\mathfrak R}{\\mathfrak S}\\left(\n\\lambda\,S_n\\right)$ 
 exhibits a commutative monoid under multiplication. Real\nsolutions $\\lam
 bda\\left({\\bf x^n}\\right)$ of skew-reciprocal equations have\nmany rema
 rkable properties: a homogeneity of the 1st order\, a duality under\ninver
 sion of variables $x_i\\to x_i^{-1}$ and function $\\lambda\\to\\lambda^{-
 1}$\,\na monotony of $\\lambda\\left({\\bf x^n}\\right)$ with respect to e
 very $x_i$ and\nothers. We find the bounds of $\\lambda\\left({\\bf x^n}\\
 right)$ which are given \nby arithmetic and harmonic means of the set $\\{
 x_1\,\\ldots\,x_n\\}$.\n
LOCATION:https://researchseminars.org/talk/CANT2022/18/
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