Related papers: Remarks on rational vector fields on $\mathbb C\ma…
In this paper, we present a generic parametrization of generically zero-dimensional parametric polynomial systems. More specifically, we study the specialization properties of the Rational Univariate Representation and derive bounds on the…
An analytic classification of generic anti-polynomial vector fields $\dot z = \overline{P(z)}$ is given in term of a topological and an analytic invariants. The number of generic strata in the parameter space is counted for each degree of…
We introduce a family of mathematical objects called $\mathcal{P}$-schemes, where $\mathcal{P}$ is a poset of subgroups of a finite group $G$. A $\mathcal{P}$-scheme is a collection of partitions of the right coset spaces $H\backslash G$,…
Orbits of families of vector fields on a subcartesian space are shown to be smooth manifolds. This allows for a global description of a smooth geometric structure on a family of manifolds in terms of a single object defined on the…
Let $k$ be a field of characteristic not equal to $2,3$, $\mathbb{O}$ an octonion over $k$ and $\mathcal{J}$ the exceptional Jordan algebra defined by $\mathbb{O}$. We consider the prehomogeneous vector space $(G,V)$ where $G=GE_6\times…
To any real rational function with generic ramification points we assign a combinatorial object, called a garden, which consists of a weighted labeled directed planar chord diagram and of a set of weighted rooted trees each corresponding to…
In this Master of Science Thesis I introduce geometric algebra both from the traditional geometric setting of vector spaces, and also from a more combinatorial view which simplifies common relations and operations. This view enables us to…
Given fields $k \subseteq L$, our results concern one parameter $L$-parametric polynomials over $k$, and their relation to generic polynomials. The former are polynomials $P(T,Y) \in k[T][Y]$ of group $G$ which parametrize all Galois…
We study degree two unirational parameterizations of geometrically rational surfaces over the real field.
We discuss various old and new definitions of the notion of a vector field on a convenient manifold that can be proved to give rise to Lie algebras, and are in finite dimensions equivalent to the standard notion of a vector field.
In this paper, a new criterion is given to determine the $p-$rationality of some complex cubic number fields in terms of $ p-$divisibility of certain terms of a third-order recurrence sequence, several illustrated examples are…
All rational parametric curves with prescribed polynomial tangent direction form a vector space. Via tangent directions with rational norm, this includes the important case of rational Pythagorean hodograph curves. We study vector subspaces…
We study the rational approximation properties of special manifolds defined by a set of polynomials with rational coefficients. Mostly we will assume the case of all polynomials to depend on only one variable. In this case the manifold can…
We classify $G$-solid rational surfaces over the field of complex numbers.
This paper describes the construction of a canonical compactification of the space of trajectories and of the unstable/stable sets of a generic gradient like vector field on a closed manifold as well as a canonical structure of a smooth…
A general type of ray class fields of global function fields is investigated. The systematic computation of their genera leads to new examples of curves over finite fields with comparatively many rational points.
This short report establishes some basic properties of smooth vector fields on product manifolds. The main results are: (i) On a product manifold there always exists a direct sum decomposition into horizontal and vertical vector fields.…
Given a geometric structure on $\mathbb{R}^{n}$ with $n$ even (e.g. Euclidean, symplectic, Minkowski, pseudo-Euclidean), we analyze the set of points inside the domain of definition of an arbitrary given $\mathcal{C}^1$ vector field, where…
In this paper, we investigate vector fields on polyhedral complexes and their associated trajectories. We study vector fields which are analogue of the gradient vector field of a function in the smooth case. Our goal is to define a nice…
On the one hand the algebras of linear operators here act on finite-dimensional vector spaces, and on the other hand the point of view is generally an analysts'. Also, one might think of algebras as being used to add more data to basic…