Related papers: Computing the decomposition group of a zero-dimens…

If K/F is a finite abelian Galois extension of global fields whose Galois group has exponent t, we prove that there exists a short exact sequence that has as a consequence that if t is square free, then Dec(K/F)=Br_{t}(K/F) which we use to…

Following Nazarov's suggestion~\cite{Naz1}, we refer to the cyclotomic Nazarov-Wenzl algebra as the cyclotomic Brauer algebra. When the cyclotomic Brauer algebra is isomorphic to the endomorphism algebra of $M_{I_i, r}$-- the tensor product…

Let $I$ be a homogeneous ideal in $R=\mathbb K[x_0,\ldots,x_n]$, such that $R/I$ is an Artinian Gorenstein ring. A famous theorem of Macaulay says that in this instance $I$ is the ideal of polynomial differential operators with constant…

Polynomial system solving arises in many application areas to model non-linear geometric properties. In such settings, polynomial systems may come with degeneration which the end-user wants to exclude from the solution set. The…

Cylindrical Algebraic Decomposition (CAD) has long been one of the most important algorithms within Symbolic Computation, as a tool to perform quantifier elimination in first order logic over the reals. More recently it is finding…

For any field $K$ and for a completely arbitrary graph $E$, we characterize the Leavitt path algebras $L_K(E)$ that are indecomposable (as a direct sum of two-sided ideals) in terms of the underlying graph. When the algebra decomposes, it…

The dominant theme of this thesis is the construction of matrix representations of finite solvable groups using a suitable system of generators. For a finite solvable group $G$ of order $N = p_{1}p_{2}\dots p_{n}$, where $p_{i}$'s are…

We study a notion of indecomposability in differential algebraic groups which is inspired by both model theory and differential algebra. After establishing some basic definitions and results, we prove an indecomposability theorem for…

In this paper, we extend the idea of comprehensive Gr\"{o}bner bases given by Weispfenning (1992) to border bases for zero dimensional parametric polynomial ideals. For this, we introduce a notion of comprehensive border bases and border…

We discuss the decomposability of torsion-free abelian groups. We show that among computable groups of finite rank this property is $\Sigma^0_3$-complete. However, when we consider groups of infinite rank, it becomes $\Sigma^1_1$-complete,…

This work provides a method(an algorithm) for solving the solvable unary algebraic equation $f(x)=0$ ($f(x)\in\mathbb{Q}[x]$) of arbitrary degree and obtaining the exact radical roots. This method requires that we know the Galois group as…

Let $K$ be a number field and $f\in K[X]$ an irreducible monic polynomial with coefficients in $O_K$, the ring of integers of $K$. We aim to enounce an effective criterion, in terms of the Galois group of $f$ over $K$ and a linear…

The computation of triangular decompositions are based on two fundamental operations: polynomial GCDs modulo regular chains and regularity test modulo saturated ideals. We propose new algorithms for these core operations relying on modular…

In this paper, we describe new methods to compute the radical (resp. real radical) of an ideal, assuming it complex (resp. real) variety is finite. The aim is to combine approaches for solving a system of polynomial equations with dual…

Call a monomial ideal M "generic" if no variable appears with the same nonzero exponent in two distinct monomial generators. Using a convex polytope first studied by Scarf, we obtain a minimal free resolution of M. Any monomial ideal M can…

We classify indecomposable summands of mixed tensor powers of the natural representation for the general linear supergroup up to isomorphism. We also give a formula for the characters of these summands in terms of composite supersymmetric…

We develop methods to study $2$-dimensional $2$-adic Galois representations $\rho$ of the absolute Galois group of a number field $K$, unramified outside a known finite set of primes $S$ of $K$, which are presented as Black Box…

We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. We propose a general algebraic framework to find the solutions and to…

We investigate the representation of symmetric polynomials as a sum of squares. Since this task is solved using semidefinite programming tools we explore the geometric, algebraic, and computational implications of the presence of discrete…

In this paper we classify solvable Leibniz algebras whose nilradical is a null-filiform algebra. We extend the obtained classification to the case when the solvable Leibniz algebra is decomposed as a direct sum of its nilradical, which is a…