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DTSTART:20260308T070000
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DTSTART;TZID=America/Toronto:20261020T100000
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URL:https://uwaterloo.ca/pure-mathematics/events/number-theory-seminar-xing
 chi-ruan-hilbert-kamke-problem
SUMMARY:Number Theory Seminar | Xingchi Ruan | The Hilbert-Kamke Problem in
 \nNumber Fields
CLASS:PUBLIC
DESCRIPTION:XINGCHI RUAN (UNIVERSITY OF WATERLOO) \n\n_The Hilbert-Kamke Pr
 oblem in Number Fields_ \nThe Hilbert-Kamke problem is a generalization of
  Waring’s problem to\na system of equations\, in which the sums of the f
 irst n powers of the\nsame variables are given. By the circle method\, the
  existence of\nsolutions depends on a positive lower bound for the singula
 r series.\nIn this talk\, we study this problem over a number field. We se
 t up the\ncircle method\, following Siegel and Wang\, and write the singul
 ar\nseries as a product of local densities. Under a necessary arithmetic\n
 condition\, and with enough variables\, we prove that the singular\nseries
  is bounded below by a positive constant depending only on the\nnumber of 
 equations and the degree of the field. The proof is given\nfirst for quadr
 atic fields and then for every number field. \nMC 5501
DTSTAMP:20261011T173826Z
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