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UID:20250714T203803EDT-59298nhvxF@132.216.98.100
DTSTAMP:20250715T003803Z
DESCRIPTION:Title:New results on the real Jacobian conjecture\n\n \n\nAbstr
 act:\n\nThe real Jacobian conjecture said:\n	If F=(f\,g):R→R be a polynomia
 l map such that det DF (x) is different from zero for all x ∈ R 2 \, then 
 F is injective.\n	\n	This conjecture had a negative answer by Pinchuk in 199
 4. Now several authors look for adding an additional assumption to the fac
 t that det(DF(x)) is different from zero for all x ∈ R2\, in order that th
 e conjecture holds. The next two theorems are proved using qualitative the
 ory of the ordinary differential equations in the plane. More precisely in
  the talk we will show how the Poincaré compactification of polynomial vec
 tor fields\, and the Poincaré-Hopf Theorem are used for proving the next t
 wo results.\n	\n	Theorem 1. Let F=(f\,g):R2→R2 be a polynomial map such that
  det(DF(x)) is different from zero for all x ∈ R2. We assume that the degr
 ees of f and g are equal and that the higher homogeneous terms of the poly
 nomials f and g do not have real linear factors in common\, then F is inje
 ctive.\n	\n	Theorem 2. Let F=(f\,g):R2→R2 be a polynomial map such that det(
 DF(x)) is different from zero for all x ∈ R2 and F(0\,0) = (0\,0). If the 
 higher homogeneous terms of the polynomials f f_x + g g_x and f f_y + g g_
 y do not have real linear factors in common\, then F is injective.\n
DTSTART:20190910T190000Z
DTEND:20190910T200000Z
LOCATION:Room 1214\, Burnside Hall\, CA\, QC\, Montreal\, H3A 0B9\, 805 rue
  Sherbrooke Ouest
SUMMARY:Jaume Llibre (Universitat Autònoma de Barcelona)
URL:/mathstat/channels/event/jaume-llibre-universitat-
 autonoma-de-barcelona-300200
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