arXiv subjects

Ibrahim Nonkane

Publications and source records attributed to Ibrahim Nonkane.

5 recordsLinked to original sources

arXiv2026

Resultant of an equivariant polynomial system with respect to a direct product of $r$ symmetric groups

In this paper we study the resultant of systems of homogeneous multivariate polynomials which are equivariant under the action of a direct product of symmetric groups. We first treat, in detail, the case of a product of two symmetric groups, and establish a decomposition formula for the resultant of such systems. We then show that this decomposition, together with the underlying combinatorics, extends to an arbitrary (finite) direct product of $r$ symmetric groups. Thanks to these decomposition formulas, we prove that the discriminant of a multivariate homogeneous polynomial invariant under a direct product of $r$ symmetric groups splits into a product of resultants of smaller size that are easier to compute.

math.AC↗

arXiv2026

Decomposition of the polynomials over the spherical subalgebra

Given a finite subgroup $W \subset \GL(\fh)$ of the linear group of a finite-dimensional complex vector field $\fh$, it is a well-studied problem to describe the structure of the symmetric algebra $B= \sym(\fh^*)$ as a representation of $G$, and also as a module over the ring of invariant differential operators under $W$ in the ring $\D(\fh)$ of differential operators on $\fh$. Since the rational Cherednik algebra $H_c(W,\fh)$ and the spherical algebra $eH_ce$ are respectively universal deformations of the ring $\D(\fh)$ and the ring $\D(\fh)^W$of $W$-invariant differential operators, we would like to build an analogy between the decomposition of modules over the invariant differential operators in \cite{ Nonk1, Nonk2, Nonk3} and the decomposition of modules over the sperical subalgebra of the rational Cherednik algebra. The ring $\D_c= eH_ce$ inherits the natural grading of $B$, and we let $\D_c^0 \subset \D_c$ and $\D_c^{-} \subset \D_c$ be subset of elements of degree 0 end strictly negative degree, respectively. Our main result is that there is for all finite reflection groups a lowest weight description of the category of $\D_c$-modules of $B$ where the ring $\Rc_c= \D_c^0 / \Dc_c^0 \cap \D_c \D_c^{-}$ plays the very important role of Cartan algebra.

math.RT↗

arXiv2025

Resultant of an equivariant polynomial system with respect to direct product of symmetric groups

In this note, we consider the resultant of systems of homogeneous multivariate polynomials which are equivariant under the action of direct product of two symmetric groups. We establish a decomposition formula for the resultant of such systems. Thanks to that decomposition formula we prove that the discriminant of an invariant multivariate homogeneous polynomial under a direct product of symmetric groups splits into smaller resultants that are easier to compute.

math.AC↗

arXiv2024

Effective Bounds on Topological Types of Real Algebraic and Semialgebraic Sets

In this paper, we revisit the problem of classifying real algebraic and semialgebraic sets by their topological types, focusing on establishing the effectiveness of bounds rather than deriving new quantitative estimates. Building on Hardt's theorem and leveraging a model-theoretic framework, we prove the existence of effective bounds on the number of distinct topological types for real algebraic sets defined by polynomials of bounded degree. While less precise than previously known doubly exponential bounds derived from methods such as Stratified Cylindrical Algebraic Decomposition, our results are obtained with minimal additional work, emphasizing simplicity and uniformity. Additionally, we establish the existence of an effective bound on the complexity of the semialgebraic homeomorphisms that describe representatives of the topological equivalence classes, offering a novel perspective on their explicit classification. These findings complement existing results by demonstrating the effectiveness of uniform bounds within a logical and geometric framework. The paper concludes with an extension of these results to semialgebraic sets of bounded complexity, further emphasizing the uniformity of the approach across varying real closed fields.

math.AG↗

arXiv2021

Higher Specht polynomials and modules over the Weyl algebra

In this paper, we study an irreducible decomposition structure of the $\Dc$-module direct image $π_+(\Oc_{ \bC^n})$ for the finite map $π: \bC^n \to \bC^n/ ({\Sc_{n_1}\times \cdots \times \Sc_{n_r}}).$ We explicitly construct the simple component of $π_+(\Oc_{\bC^n})$ by providing their generators and their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a $\D$-module decomposition of the polynomial ring localized at the discriminant of $π$. Furthermore, we study the action invariants, differential operators, on the higher Specht polynomials.

math.AG↗