.. .. Alternating, 55 1.4 Basic notions about graphs and trees, p.61

. .. Snake-trees, 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

?. ·-·-·-?-a-k-=-b-1-b-2-·-·-·-b-l, 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)

?. Second, 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

D. Charalambos, K. C. Aliprantis, and . Border, Infinite dimensional analysis. Third. A hitchhiker's guide, vol.122, p.71, 2006.

C. Adams and R. Franzosa, Introduction to Topology: Pure and Applied, Person Education, Inc, publishing, p.64, 2008.

S. Vladimir-igorevich-arnold, A. N. Medgidovich-gusein-zade, and . Varchenko, Modern Birkhäuser Classics. Classification of critical points, caustics and wave fronts, Translated from the Russian by Ian Porteous based on a previous translation by Mark Reynolds, vol.1, p.169, 2012.

M. Albert, C. Homberger, and J. Pantone, Equipopularity classes in the separable permutations, In: Electron. J. Combin, vol.22, 2015.

D. André, Développements de sec x et de tang x, In: CR Acad. Sci. Paris, vol.88, p.49, 1879.

D. André, Sur les permutations alternées, Journal de mathématiques pures et appliquées 7 (1881), p.49

A. Vladimir-igorevich, Nombres d'Euler, de Bernoulli et de Springer pour les groupes de Coxeter et les espaces de morsification : le calcul des serpents, pp.61-98, 2000.

A. Vladimir-igorevich, Snake calculus and the combinatorics of the Bernoulli, Euler and Springer numbers of Coxeter groups, Uspekhi Mat. Nauk, vol.47, pp.3-45, 1992.

F. Bassino, The Brownian limit of separable permutations, Ann. Probab, vol.46, pp.2134-2189, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01275310

P. Bose, J. F. Buss, and A. Lubiw, Pattern matching for permutations, Inform. Process. Lett, vol.65, pp.277-283, 1998.

M. Berger, I. Geometry, and . Universitext, Translated from the 1977 French orig

. Springer-verlag, , p.164, 2009.

E. Brieskorn and H. Knörrer, Plane algebraic curves, p.721, 1986.

J. , A. Bondy, and U. Murty, Graduate Texts in Mathematics, vol.244, p.61, 2008.

T. S. Bolis, Degenerate critical points, In: Math. Mag, vol.53, p.119, 1980.

B. Bollobás, Graduate Texts in Mathematics, Modern graph theory, vol.184, p.61, 1998.

M. Bouvel and D. Rossin, The longest common pattern problem for two permutations, Pure Math. Appl. (PU.M.A.), vol.17, issue.2, p.99, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00115598

R. Castellini, The topology of A'Campo deformations of singularities: an approach through the lotus, p.136, 2015.
URL : https://hal.archives-ouvertes.fr/tel-01207005

J. Lowell-coolidge, A treatise on algebraic plane curves, p.165, 1959.

M. Coste, An introduction to semialgebraic geometry, RAAG network school, vol.145, p.128, 2002.

M. Coste, Real algebraic sets, Notes de cours, vol.129, p.128, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00368053

M. Coste and M. Puente, Atypical values at infinity of a polynomial function on the real plane: an erratum, and an algorithmic criterion, J. Pure Appl. Algebra, vol.162, pp.111-111, 2001.

C. Davis, Extrema of a polynomial, In: Amer. Math. Monthly, vol.64, pp.55-57, 1957.

, Graduate Texts in Mathematics, vol.173, p.61, 2010.

A. Douady, Géométrie dans les espaces de paramètres, IREM, Cahiers rouges, 23, vol.2, 1997.

D. Eisenbud, With a view toward algebraic geometry, Graduate Texts in Mathematics, vol.150, p.137, 1995.

J. Ferrera-cuesta and M. Puente, The asymptotic values of a polynomial function on the real plane, J. Pure Appl. Algebra, vol.106, issue.95, p.136, 1996.

G. Fischer, Student Mathematical Library. Translated from the 1994 German original by Leslie Kay, American Mathematical Society, vol.15, pp.164-166, 2001.

H. Flanders, Differentiation under the integral sign, In: Amer. Math. Monthly, vol.80, p.59, 1973.

P. G. Frè and . Gravity, Development of the theory and basic physical applications, vol.1, p.125, 2013.

R. Evelia, . García, and . Barroso, Sur les courbes polaires d'une courbe plane réduite", In: Proc. London Math. Soc, vol.81, issue.3, pp.1-28, 2000.

R. Evelia, . García, and . Barroso, Invariants des singularités de courbes planes et courbure des fibres de Milnor, vol.36, 1996.

R. Evelia, . García, and . Barroso, Un théorème de décomposition pour les polaires génériques d'une courbe plane, C. R. Acad. Sci. Paris Sér. I Math, vol.326, pp.82713-82722, 1998.

E. R. García-barroso, P. Pérez, and P. Popescu-pampu, Greuel and S. Xambo-Descamps. Festschrift for Antonio Campillo on the occasion of his 65th birthday (cit, Singularities, algebraic geometry, commutative algebra and related topics, vol.77, p.66, 2018.

É. Ghys, Intersecting curves (variation on an observation of Maxim Kontsevich), In: Amer. Math. Monthly, vol.120, p.77, 0190.

É. Ghys, A singular mathematical promenade, ENS Éditions, vol.22, p.186, 2017.

G. Thomas and G. , Positive definite matrices and Sylvester's criterion, In: Amer. Math. Monthly, vol.98, p.118, 1991.

M. M. Izrail-moiseevich-gelfand, A. V. Kapranov, and . Zelevinsky, Discriminants, resultants, and multidimensional determinants. Mathematics: Theory & Applications, p.143, 1994.

É. Ghys and J. Leys, Le pli et la fronce-un théorème de Whitney, p.169, 2009.

E. Gossett, Discrete mathematics with proof, pp.61-64, 2009.

V. Guillemin and A. Pollack, Differential topology, N.J, p.132, 1974.

J. M. Johnson and J. Kollár, How small can a polynomial be near infinity?, In: Amer. Math. Monthly, vol.118, p.119, 2011.

M. M. Kapranov, The permutoassociahedron, Mac Lane's coherence theorem and asymptotic zones for the KZ equation, J. Pure Appl. Algebra, vol.85, p.77, 1993.

S. Kitaev, Patterns in permutations and words, vol.32, pp.95-97, 2011.

D. E. Knuth, Fasc. 1. MMIX, a RISC computer for the new millennium, vol.1, p.134, 2005.

. Steven-george-krantz, Handbook of complex variables, p.131, 1999.

P. Vladimir, B. Z. Kostov, and . Shapiro, On arrangements of roots for a real hyperbolic polynomial and its derivatives, Bull. Sci. Math, vol.126, p.53, 2002.

S. K. Lando, Student Mathematical Library. Translated from the 2002 Russian original by the author, Lectures on generating functions, vol.23, pp.53-55, 2003.

L. Libkin and V. Gurvich, Trees as semilattices, Discrete Math, vol.145, p.80, 1995.

D. Lê, F. Michel, and C. Weber, Sur le comportement des polaires associées aux germes de courbes planes, Compositio Math. 72, vol.1, pp.87-113, 1989.

Y. Matsumoto, An introduction to Morse theory, Translations of Mathematical Monographs. Translated from the 1997 Japanese original by Kiki Hudson and Masahico Saito, Iwanami Series in Modern Mathematics, vol.208, p.124, 2002.

H. Maugendre, Topologie comparée d'une courbe polaire et de sa courbe discriminante, In: Rev. Mat. Complut, vol.12, pp.439-450, 1999.

J. Willard-milnor, Topology from the differentiable viewpoint, The University Press of, p.132, 1965.

J. Willard-milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, issue.61, p.122, 1968.

J. Mycielski, Mathematical Notes: Polynomials with Preassigned Values at their Branching Points, In: Amer. Math. Monthly, vol.77, pp.853-855, 1970.

H. Osborn, Pure and Applied Mathematics. Foundations and Stiefel Whitney classes, vol.1, p.126, 1982.

J. Plücker, Sur les points singuliers des courbes, Journal de Mathématiques pures et appliquées, vol.2, pp.11-15, 1837.

H. Poincaré, History of Mathematics. Rendiconti del Circolo Matematico di Palermo, vol.37, p.228, 1904.

J. Poncelet, Théorie des polaires réciproques". In: Annales de Mathématiques pures et appliqués 8 (1817), pp.201-232

P. Popescu-pampu, Arbres de contact des singularités quasi-ordinaires et graphes d'adjacence pour les 3-variétés réelles, p.61, 2001.

G. Reeb, Sur les points singuliers d'une forme de Pfaff complètement intégrable ou d'une fonction numérique, vol.222, pp.847-849, 1946.

G. Victor-reklaitis, A Wiley-Interscience Publication. Methods and applications, vol.118, p.117, 1983.

J. Serre, , p.62, 1980.

N. Shafiei, Recursive Definitions and Structural Induction, p.65

W. Louis, R. A. Shapiro, and . Sulanke, Bijections for the Schröder numbers, In: Math. Mag, vol.73, p.189, 2000.

E. Shult and D. Surowski, Algebra-a teaching and source book, vol.83, p.80, 2015.

R. Peter and S. , A survey of alternating permutations, Combinatorics and graphs, vol.531, pp.165-196, 2010.

R. Peter and S. , Cambridge Studies in Advanced Mathematics, Enumerative combinatorics, vol.1, p.64, 2012.

R. Peter and S. , Catalan numbers, p.189, 2015.

R. Peter-stanley-;-hipparchus, S. Plutarch, and H. , In: Amer. Math. Monthly, vol.104, p.189, 1997.

D. Daniel, R. E. Sleator, W. Tarjan, and . Thurston, Rotation distance, triangulations, and hyperbolic geometry, J. Amer. Math. Soc, vol.1, issue.3, p.187, 1988.

B. Teissier, Variétés polaires. I. Invariants polaires des singularités d'hypersurfaces, In: Invent. Math, vol.40, pp.267-292, 1977.

B. Teissier, Quelques points de l'histoire des variétés polaires, de Poncelet à nos jours, Séminaire d'Analyse, vol.II, 1987.

B. Teissier and A. G. Flores, Local polar varieties in the geometric study of singularities, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01348838

R. J. Trudeau, Introduction to graph theory. Corrected reprint of the 1976 original, vol.63, p.61, 1993.

W. Thomas-tutte, Graph theory, Encyclopedia of Mathematics and its Applications, vol.21, p.61, 1984.

S. Vickers, Cambridge Tracts in Theoretical Computer Science, vol.5, p.79, 1989.

O. Y. Viro, Some integral calculus based on Euler characteristic, Topology and geometry-Rohlin Seminar, vol.1346

. Springer, , p.128, 1988.

C. Wall, Singular points of plane curves, vol.63, p.143, 2004.

C. Wall, A geometric introduction to topology, p.131, 1972.

R. Walker, Algebraic curves. Reprint of the 1950 edition, vol.117, p.116, 1978.

J. West, Generating trees and the Catalan and Schröder numbers, Discrete Math, vol.146, p.189, 1995.

S. Willard, General topology, p.71, 1970.

. Frederick-shenstone-woods, Advanced calculus: a course arranged with special reference to the needs of students of applied mathematics, p.59, 1926.