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# Combinatorics of singularities of some 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 $\mathbb{K}[[x_1,...,x_e]][y]$ and by a preliminary change of variables we may assume that the leading term of the discriminant of $f$ contains a power of $x_1$. After a monomial transformation we get a quasi-ordinary polynomial with a root in $\mathbb{K}[[x_{1}^{\frac{1}{n}},...,x_{e}^{\frac{1}{n}}]]$ for some $n\in{\mathbb N}$. By taking the preimage of $f$ we get a solution $y\in\mathbb{K}_{C}[[x_{1}^{\frac{1}{n}},...,x_{e}^{\frac{1}{n}}]]$ of $f(x_1,...,x_e,y) = 0$, where $\mathbb{K}_{C}[[x_{1}^{\frac{1}{n}},...,x_{e}^{\frac{1}{n}}]]$ 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 $M = F_1 A + \ldots + F_r A$ where $A=\mathbb{K}[f_{1},...,f_{s}]$ and $F_1,\ldots, F_r \in \mathbb{K}[t]$. 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 their invariants, using the module of K\"ahler differentials.
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Theses

Cited literature [31 references]

https://hal.archives-ouvertes.fr/tel-01616702
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Submitted on : Saturday, October 14, 2017 - 2:01:52 AM
Last modification on : Wednesday, November 3, 2021 - 9:18:32 AM
Long-term archiving on: : Monday, January 15, 2018 - 12:12:21 PM

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Thesis.pdf
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• HAL Id : tel-01616702, version 1

### Citation

Ali Abbas. Combinatorics of singularities of some curves and hypersurfaces. Algebraic Geometry [math.AG]. Université d'Angers, 2017. English. ⟨tel-01616702⟩

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