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Combinatorics of singularities of some special curves and hypersurfaces

Abstract : The thesis is made up of two parts. In the first part we generalize the Abhyankar-Moh theory to a special kind of polynomials, called free polynomials. We take a polynomial f in K[[x1, ..., xe]][y] and by a preliminary change of variables we may assume that the leading term of the discriminant of f contains a power of x1.After a monomial transformation we get a quasi-ordinary polynomial with a root in K[[x1n1 , ..., x1ne ]] for some n ∈ N. By taking the preimage of f we get asolution y ∈ KC[[x1n1 , ..., x1ne ]] of f(x1, ..., xe, y) = 0,where KC[[x1n1 , ..., x1ne ]] is the ring of formal fractional power series with support in a specific line free cone C. Then we construct the set of characteristic exponents of y, and we generalize some of the results concerning quasi-ordinary polynomials to f. In the second part, we give a procedure to calculate the monoid of degrees of the module M = F1A + . . . + FrA where A = K[f1, ..., fs] andF1, . . . , Fr ∈ K[t]. Then we give some applications to the problem of the classification of plane polynomial curves (that is, plane algebraic curves parametrized by polynomials) with respect to some of the irinvariants, using the module of Kähler differentials.
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Submitted on : Thursday, May 27, 2021 - 3:16:08 PM
Last modification on : Wednesday, November 3, 2021 - 9:18:27 AM
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  • HAL Id : tel-03239539, version 1


Ali Abbas. Combinatorics of singularities of some special curves and hypersurfaces. Discrete Mathematics [cs.DM]. Université d'Angers, 2017. English. ⟨NNT : 2017ANGE0098⟩. ⟨tel-03239539⟩



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