Related papers: Determination and (re)parametrization of rational …
Parametric Gr\"obner bases have been studied for more than 15 years and are now a further developed subject. Here we propose a general study of parametric standard bases, that is with local orders. We mainly focus on the commutative case…
We use function field analytic number theory to establish the irreducibility and dimension of the moduli space that parameterises morphisms of fixed degree from $\mathbb{P}^2$ to an arbitrary smooth hypersurface of sufficiently small…
Motivated by applications in CNC machining, we provide a characterization of surfaces which are enveloped by a one-parametric family of congruent rotational cones. As limit cases, we also address developable surfaces and ruled surfaces. The…
We establish a couple of dynamical properties of surjective rational maps $f: X \dashrightarrow X$ for smooth projective surfaces $X$. We also give a numerical characterization of regular $f$ in the case when $X$ is a del Pezzo surface.…
We present a discrete theory for modeling developable surfaces as quadrilateral meshes satisfying simple angle constraints. The basis of our model is a lesser known characterization of developable surfaces as manifolds that can be…
The field of numerical algebraic geometry consists of algorithms for numerically solving systems of polynomial equations. When the system is exact, such as having rational coefficients, the solution set is well-defined. However, for a…
Affine rotation surfaces are a generalization of the well-known surfaces of revolution. Affine rotation surfaces arise naturally within the framework of affine differential geometry, a field started by Blaschke in the first decades of the…
In this paper we show that not all affine rational complex surfaces can be parametrized birationally and surjectively. For this purpose, we prove that, if S is an affine complex surface whose projective closure is smooth, a necessary…
Surface parameterization is a fundamental concept in fields such as differential geometry and computer graphics. It involves mapping a surface in three-dimensional space onto a two-dimensional parameter space. This process allows for the…
We give an upper bound for the degree of rational curves in a family that covers a given birational ruled surface in projective space. The upper bound is stated in terms of the degree, sectional genus and arithmetic genus of the surface. We…
For a finite dimensional vector space equipped with a $\mathbb C$-algebra structure, one can define rational maps using the algebraic structure. In this paper, we describe the growth of the degree sequences for this type of rational maps.
Prediction capability is considered an important issue in response surface methodology. Following the line of argument that a design should have several desirable properties we have extended an existing compound design criterion to include…
We classify geometrically integral regular del Pezzo surfaces which are not geometrically normal over imperfect fields of positive characteristic. Based on this classification, we show that a three-dimensional terminal del Pezzo fibration…
In this article, a new method for segmentation and restoration of images on two-dimensional surfaces is given. Active contour models for image segmentation are extended to images on surfaces. The evolving curves on the surfaces are…
Comessatti proved that the set of real points of a rational real algebraic surface is either a nonorientable surface, or the two-sphere, or the torus. Conversely, it is easy to see that all of these surfaces admit a rational real algebraic…
We investigate topological descriptors for 3D surface analysis, i.e. the classification of surfaces according to their geometric fine structure. On a dataset of high-resolution 3D surface reconstructions we compute persistence diagrams for…
In a previous paper, we provided some update in the treatment of the finiteness theorem for rational maps of finite degree from a fixed variety to varieties of general type. In the present paper we present another improvement, introducing…
This is an expository paper which presents the holomorphic classification of rational complex surfaces from a simple and intuitive point of view, which is not found in the literature. Our approach is to compare this classification with the…
Computational topology is an area that revisits topological problems from an algorithmic point of view, and develops topological tools for improved algorithms. We survey results in computational topology that are concerned with graphs drawn…
The purpose of this paper is to explore the question "to what extent could we produce formal, machine-verifiable, proofs in real algebraic geometry?" The question has been asked before but as yet the leading algorithms for answering such…