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Abdallah Assi

Publications and source records attributed to Abdallah Assi.

17 recordsLinked to original sources
arXiv2025

Geometry of Supermanifolds through Sheaf and Ringed Space Methods

This paper introduces the concept of supermanifolds, viewed as the super-analogues of classical manifolds. Instead of treating supermanifolds as sets of points, we adopt an algebraic-geometric perspective, emphasizing the algebra of functions and utilizing the framework of ringed spaces and sheaf theory. We begin by constructing presheaves and sheaves to define locally ringed spaces, which model the local structure of supermanifolds. Superfunctions, a key element, are shown to differ significantly from ordinary functions, leading to a richer structure. We also prove a key characterization theorem for morphisms between supermanifolds and demonstrate that the category of supermanifolds admits finite products. This approach provides a solid foundation for further studies in mathematics and theoretical physics.

math.AG↗
arXiv2021

New log-majorization results concerning eigenvalues and singular values and a complement of a norm inequality

The purpose of this paper is to establish new log-majorization results concerning eigenvalues and singular values which generalize some previous work related to a conjecture and an open question which were presented by R. Lemos and G. Soares in \cite{lemos}. In addition, we present a complement of a unitarily invariant norm inequality which was conjectured by R. Bhatia, Y. Lim and T. Yamazaki in \cite{Bhatia2}, and recently proved by T.H. Dinh, R. Dumitru and J.A. Franco in \cite{Dinh} for the Schatten p-norm with $1\leq p\leq \infty$.

math.FA↗
arXiv2021

Semigroup associated with a free polynomial

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and let $\mathbb{K}_{C}[[x_{1},...,x_{e}]]$ be the ring of formal power series in several variables with exponents in a line free cone $C$. We consider irreducible polynomials $f=y^n+a_1(\underline{x})y^{n-1}+\ldots+a_n(\underline{x})$ in $\mathbb{K}_{C}[[x_{1},...,x_{e}]][y]$ whose roots are in $\mathbb{K}_{C}[[x_{1}^{\frac{1}{n}},...,x_{e}^{\frac{1}{n}}]]$. We generalize to these polynomials the theory of Abhyankar-Moh. In particular we associate with any such polynomial its set of characteristic exponents and its semigroup of values. We also prove that the set of values can be obtained using the set of approximate roots. We finally prove that polynomials of ${\mathbb K}[[\underline{x}]][y]$ fit in the above set for a specific line free cone (see Section 4).

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arXiv2021

On canonical bases of a formal ${\mathbb K}-algebra

We study canonical bases of a subalgebra ${\bf A}={\mathbb K}[\![f_1,\dots,f_s]\!]\subseteq {\mathbb K}[\![x_1,\dots,x_n]\!]$ over a field ${\mathbb K}$, and we associate with ${\bf A}$ a fan called the canonical fan of $\bf A$. This generalizes the notion of the standard fan of an ideal.

math.RA↗
arXiv2019

The plane Jacobian conjecture for rational curves

Let K be an algebraically closed field of characteristic zero and let f(x,y) be a nonzero polynomial of K[x,y]. We prove that if the generic element of the family $(f-λ)\_λ$ is a rational polynomial, and if the Jacobian J(f,g) is a nonzero constant for some polynomial g in K[x,y], then K[f,g] =K[x,y].

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arXiv2016

Bases of subalgebras of K[[x]] and K[x]

Let $f\_1,\ldots, f\_s$ be formal power series (respectively polynomials) in thevariable $x$. We study the semigroup of orders of the formal series inthe algebra $K[[ f1,\ldots, f\_s]] \subseteq K[[ x ]]$ (respectively the semigroup of degrees of polynomials in$K[f\_1,\ldots,f\_s]\subseteq K[x]$). We give procedures to compute thesesemigroups and several applications.

math.AG↗
arXiv2014

Numerical semigroups and applications

The aim of this manuscript is to give some basic notions related to numerical semigroups, and from these on the one hand describe a classical application to the study of singularities of plane algebraic curves, and on the other, show how numerical semigroups can be used to obtain handy examples of nonunique factorization invariants.

math.AG↗
arXiv2014

On curves with one place at infinity

Let $f$ be a plane curve. We give a procedure based on Abhyankar's approximate roots to detect if it has a single place at infinity, and if so construct its associated $δ$-sequence, and consequently its value semigroup. Also for fixed genus (equivalently Frobenius number) we construct all $δ$-sequences generating numerical semigroups with this given genus. For a $δ$-sequence we present a procedure to construct all curves having this associated sequence. We also study the embeddings of such curves in the plane. In particular, we prove that polynomial curves might not have a unique embedding.

math.AG↗
arXiv2013

Frobenius vectors, Hilbert series and gluings

Let $S_1$ and $S_2$ be two affine semigroups and let $S$ be the gluing of $S_1$ and $S_2$. Several invariants of $S$ are then related to those of $S_1$ and $S_2$; we review some of the most important properties preserved under gluings. The aim of this paper is to prove that this is the case for the Frobenius vector and the Hilbert series. Applications to complete intersection affine semigroups are also given.

math.AC↗
arXiv2013

Rational curves with one place at infinity

Let K be an algebraically closed field of characteristic zero. Given a polynomial f(x,y) in K[x,y] with one place at infinity, we prove that either f is equivalent to a coordinate, or the family (f+c) has at most two rational elements. When (f+c) has two rational elements, we give a description of the singularities of these elements.

math.AG↗
arXiv2013

The tree model of a meromorphic plane curve

We associate with a plane meromorphic curve f a tree model T(f) based on its contact structure. Then we give a description of the y-derivative of f (resp. the Jacobien J(f,g)) in terms of T(f) (resp. T(fg)). We also characterize the regularity of f in terms of its tree and we give a bound for the number of its irregular values.

math.AG↗
arXiv2013

Constructing the set of complete intersection numerical semigroups with a given Frobenius number

Delorme suggested that the set of all complete intersection numerical semigroups can be computed recursively. We have implemented this algorithm, and particularized it to several subfamilies of this class of numerical semigroups: free and telescopic numerical semigroups, and numerical semigroups associated to an irreducible plane curve singularity. The recursive nature of this procedure allows us to give bounds for the embedding dimension and for the minimal generators of a semigroup in any of these families.

math.CO↗
arXiv2009

The embedding conjecture for quasi-ordinary hypersurfaces

This paper has two objectives: we first generalize the theory of Abhyankar-Moh to quasi-ordinary polynomials, then we use the notion of approximate roots and that of generalized Newton polygons in order to prove the embedding conjecture for this class of polynomials. This conjecture -made by S.S. Abhyankar and A. Sathaye- says that if a hypersurface of the affine space is isomorphic to a coordinate, then it is equivalent to it.

math.AG↗
arXiv2005

Effective construction of irreducible curve singularities

We can associate with any irreducible curve singularity (ics) a numerical semigroup. Two ics are said to be equisingular if they have the same semigroup. Two equisingular ics have the same Milnor number. Conversely, The set of ics with a given Milnor number is a union of equisingular classes. Here we study ics from an algorithmic viewpoint, by using the notion of approximate roots. We give two algorithms: the first one constructs the canonical equation of a curve with a given semigroup. The second one gives the set of semigroups with a fixed Milnor number. The paper is backed by Maple and Mathematica programs which are available upon request.

math.AG↗
arXiv2005

On quasihomogeneous curves

A hypersurface is said to be quasihomogeneous if in suitable coordinates with assigned weights, its equation becomes weighted homogeneous in its variables. For an irreducible quasihomogeneous plane curve, the equation necessarily becomes a two term equation of the form $aY^n+bX^m$ where $n,m$ are necessarily coprime. Zariski, in a short paper, established a criterion for an algebroid curve to be quasihomogeneous and a celebrated theorem of Lin and Zaidenberg gives a global criterion for quasihomogeneity. The Lin-Zaidenberg theorem does not have a simple proof, despite having three different proofs using function theory, topology and algebraic surface theory respectively. We give here a global version of the Zariski result. As a consequence we give a proof of a slightly weaker version of the Lin-Zaidenberg Theorem, namely that a rational curve with one place at infinity is unibranch and locally quasihomogeneous if and only if it is globally quasihomogeneous, provided the ground field is algebraically closed of characteristic zero. Our method of proof leads to some interesting questions about the change in the module of differentials when we go to the integral closure.

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