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SUMMARY:Lior Bary-Soroker (Tel Aviv University)
DTSTART:20260708T150000Z
DTEND:20260708T160000Z
DTSTAMP:20260707T154745Z
UID:LNTS/200
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/LNTS/200/">G
 eneric Manin–Mumford</a>\nby Lior Bary-Soroker (Tel Aviv University) as 
 part of London number theory seminar\n\nLecture held in King's College Lon
 don\, Strand campus\, Room S3.30.\n\nAbstract\nThe unlikely intersection\,
  or Manin–Mumford\, principle asserts that an algebraic variety containi
 ng many special points must itself contain special subvarieties. An elemen
 tary illustration is the following theorem of Ihara\, Serre\, and Tate: if
  a polynomial equation $F(X\,Y)=0$ has infinitely many solutions in roots 
 of unity\, then either $X^rY^s-w$ or $X^r-w Y^s$ divides $F$\, where $w$ i
 s a root of unity. This may be viewed as the first instance of the general
  Manin–Mumford theorem describing subvarieties of semi-abelian varieties
  containing many torsion points.\n\nMotivated by recent work of Binyamini\
 , Kiro\, and Pila\, we ask: what happens if one replaces the roots of unit
 y by a different collection of algebraic numbers?\n\nI will discuss a gene
 ral criterion guaranteeing a Manin–Mumford type theorem for a set $S$ of
  algebraic numbers. Roughly speaking\, if the elements of $S$ are sufficie
 ntly generic in the Galois-theoretic sense\, then any algebraic variety co
 ntaining many points of $S^n$ must contain an obvious special subvariety\,
  cut out by equations of the form $x_i=x_j$ or $x_i=a$ with $a$ in $S$. As
  applications\, we recover a recent theorem of Binyamini\, Kiro\, and Pila
  on roots of Laguerre polynomials\, obtain analogous results for several o
 ther classical polynomial families\, and prove that the phenomenon holds a
 lmost surely for roots of random polynomials.\n
LOCATION:https://researchseminars.org/talk/LNTS/200/
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