55 1.4 Basic notions about graphs and trees, p.61 ,
66 1.6 The Poincaré-Reeb tree of a one variable polynomial, p.70 ,
74 1.8 A construction of all the separable Poincaré-Reeb trees in one variable 95 ,
where is either ? or , A i is ?-indecomposable, for i = 1,. .. , k and B j is-indecomposable, for j = 1, First step: let us prove that ? cannot be both ?-decomposable and-decomposable ,
, From the first decomposition of ?, we have ? = A 1 ? A 2 ? · · · ? A k , thus ?(1) < ?(n)
Let us suppose that #B 1 < k 1. Then ?(k 1 ) must belong to the block B 2. Since between B 1 and B 2 there is ? sign we obtain ?(1) < ?(k 1 ). Contradiction. Analogously we prove that we cannot have #B 1 > k 1 .-Second case: k 1 = 1, namely it is reduced to one element: A 1 =. Since ? = A 1 ? A 2 ? · · · ? A k , for any j ? {2,. .. , n} we have ?(j) > ?(1), we prove that #A 1 = #B 1. Denote by k 1 := #A 1 .-First case: k 1 ? 2. Since A 1 is-decomposable, because it is not ?-decomposable, we obtain ?(1) > ?(k 1 ) ,
,n we can prove as before that #A 2 = #B 2. Recursively, we obtain #A i = #B i , for i = 1, ? By taking the restriction of ? to the indices k 1 + 1 ,
, ? Recursively we apply the steps above to prove that the decomposition of A i is the same as the decomposition of B i , for i = 1
, There is a bijection between the set of the separable generic rooted transversal trees and the quotient space of binary separating trees with respect to the flop equivalence
, 13, they provide a unique decomposition (up to associativity) which corresponds to a unique separable snake. By Proposition 1.69, this separable snake corresponds to a unique separable Poincaré-Reeb tree. ? Given a separable Poincaré-Reeb tree, by Proposition 1.69, it corresponds to a unique separable snake, Given any two binary separating trees that are flop-equivalent
, One of our main problems is to realise pairs (negative-positive) of (separable)
, Poincaré-Reeb trees in two variables by a strict local minimum of a polynomial function at the origin. In Chapter 1 the binary separating trees were used as contact trees in our construction of Poincaré-Reeb trees for one variable polynomials
, There exist more complicated examples, where using a single decomposition is not enough. Namely, each tree s ? Flop(?) may provide new negative Poincaré-Reeb trees
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