NON REALITY AND NON CONNECTIVITY OF

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Niveau: Supérieur, Doctorat, Bac+8
NON REALITY AND NON CONNECTIVITY OF COMPLEX POLYNOMIALS ARNAUD BODIN Abstra t. Using the same method we provide negative answers to the following questions: Is it possible to nd real equations for omplex poly- nomials in two variables up to topologi al equivalen e (Lee Rudolph)? Can two topologi ally equivalent polynomials be onne ted by a ontin- uous family of topologi ally equivalent polynomials? 1. Introdu tion Two polynomials f; g 2 C [x; y? are topologi ally equivalent, and we will denote f g, if there exist homeomorphisms : C 2 ! C 2 and : C ! C su h that gÆ = Æf . They are algebrai ally equivalent, and we will denote f g, if we have 2 Aut C 2 and = id. It is always possible to nd real equations for germs of plane urves up to topologi al equivalen e. In fa t the proof is as follows: the topologi al type of a germ of plane urve (C; 0) is determined by the hara teristi pairs of the Puiseux expansions of the irredu ible bran hes and by the interse tion multipli ities between these bran hes. Then we an hoose the oeÆ ients of the Puiseux expansions in R (even in Z).

  • tion

  • then

  • plane urve

  • ee tive

  • ally equivalent sin

  • urves up

  • ally equivalent

  • has degree


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REALITY
AND
NON
giv
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CONNECTIVITY
umerical
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olynomial
BODIN
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