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SUMMARY:Alina Ostafe (The University of New South Wales)
DTSTART:20211208T110000Z
DTEND:20211208T120000Z
DTSTAMP:20260423T021351Z
UID:hnts/45
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/hnts/45/">Mu
 ltiplicative and additive relations for values of rational functions and p
 oints on elliptic curves</a>\nby Alina Ostafe (The University of New South
  Wales) as part of Heilbronn number theory seminar\n\n\nAbstract\nFor give
 n rational functions $f_1\,\\ldots\,f_s$ defined over a number field\, Bom
 bieri\, Masser and Zannier (1999) proved that the algebraic numbers $\\alp
 ha$ for which the values $f_1(\\alpha)\,\\ldots\,f_s(\\alpha)$ are multipl
 icatively dependent are of bounded height (unless this is false for an obv
 ious reason).\n\nMotivated by this\, we present various extensions and rec
 ent finiteness results on multiplicative relations of values of rational f
 unctions\, both in zero and positive characteristics. In particular\, one 
 of our results shows that\, given non-zero rational functions $f_1\,\\ldot
 s\,f_m\,g_1\,\\ldots\,g_n \\in \\mathbb{Q}(X)$ and an elliptic curve $E$ d
 efined over $\\mathbb{Q}$\, for any sufficiently large prime $p$\, for all
  but finitely many $\\alpha\\in\\overline{\\mathbb{F}}_p$\, at most one of
  the following two can happen: $f_1(\\alpha)\,\\ldots\,f_m(\\alpha)$ satis
 fy a short multiplicative relation or the points $(g_1(\\alpha)\,\\cdot)\,
  \\ldots\,(g_n(\\alpha)\,\\cdot)\\in E_p$ satisfy a short linear relation 
 on the reduction $E_p$ of $E$ modulo $p$.\n
LOCATION:https://researchseminars.org/talk/hnts/45/
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