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SUMMARY:Michael Filaseta
DTSTART:20200720T160000Z
DTEND:20200720T170000Z
DTSTAMP:20260423T021216Z
UID:UNYONTC/4
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/UNYONTC/4/">
 On a dense universal Hilbert set</a>\nby Michael Filaseta as part of Upsta
 te New York Online Number Theory Colloquium\n\n\nAbstract\nA $universal\\ 
  Hilbert\\  set$\nis an infinite set $\\mathcal S \\subseteq \\mathbb Z$\n
 having the property that for every $F(x\,y) \\in \\mathbb Z[x\,y]$\nwhich 
 is irreducible in $\\mathbb Q[x\,y]$ and satisfies $\\deg_{x} (F) \\ge 1$\
 ,   \nwe have that for all but finitely many $y_{0} \\in \\mathcal S$\, th
 e polynomial \n$F(x\,y_{0})$ is irreducible in $\\mathbb Q[x]$.  \nThe exi
 stence of universal Hilbert sets is due to P.C. Gilmore and A. Robinson in
  1955\,\nand since then a number of explicit examples have been given.  \n
 Universal Hilbert sets of density $1$ in the integers have been shown to e
 xist \nby Y. Bilu in 1996 and P. D\\`ebes and U. Zannier in 1998.\nIn this
  talk\, we discuss a connection between universal Hilbert sets and \nSiege
 l's Lemma on the finiteness of integral points on a curve\nof genus $\\ge 
 1$\, and explain how a result of K.Ford (2008) implies\nthe existence of a
  universal Hilbert set $\\mathcal S$ satisfying\n\\[\n|\\{ m \\in \\mathbb
  Z: m \\not\\in \\mathcal S\, |m| \\le X \\}| \\ll \\dfrac{X}{(\\log X)^{\
 \delta}}\,\n\\]\nwhere $\\delta = 1 - (1+\\log\\log 2)/(\\log 2) = 0.08607
 1\\ldots$.  This is joint work with Robert Wilcox.\n
LOCATION:https://researchseminars.org/talk/UNYONTC/4/
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