相关论文: Enumerative Real Algebraic Geometry

The relationship between algebraic geometry and the inferential framework of the Bayesian Networks with hidden variables has now been fruitfully explored and exploited by a number of authors. More recently the algebraic formulation of…

This article reviews some aspects in the current relationship between mathematical and numerical General Relativity. Focus is placed on the description of isolated systems, with a particular emphasis on recent developments in the study of…

We associate a half-integer number, called {\em the quantum index}, to algebraic curves in the real plane satisfying to certain conditions. The area encompassed by the logarithmic image of such curves is equal to $\pi^2$ times the quantum…

In this survey article, we present interactions between algebraic geometry and computer vision, which have recently come under the header of algebraic vision. The subject has given new insights in multiple view geometry and its application…

By recasting metrical geometry in a purely algebraic setting, both Euclidean and non-Euclidean geometries can be studied over a general field with an arbitrary quadratic form. Both an affine and a projective version of this new theory are…

Since its introduction by Gauss, Matrix Algebra has facilitated understanding of scientific problems, hiding distracting details and finding more elegant and efficient ways of computational solving. Today's largest problems, which often…

For a smooth projective curve, the cycles of subordinate or, more generally, secant divisors to a given linear series are among some of the most studied objects in classical enumerative geometry. We consider the intersection of two such…

The Numerical Assembly Technique is extended to investigate arbitrary planar frame structures with the focus on the computation of natural frequencies. This allows us to obtain highly accurate results without resorting to spatial…

Recently, a geometrical characterization of vector spaces served to generalize them into a new class of algebras. Instead of the algebraic properties of the underlying fields, we generalized the recently discovered property of such spaces…

The paper gives an overview of recent advances in structural equation modeling. A structural equation model is a multivariate statistical model that is determined by a mixed graph, also known as a path diagram. Our focus is on the…

The image of a polynomial map is a constructible set. While computing its closure is standard in computer algebra systems, a procedure for computing the constructible set itself is not. We provide a new algorithm, based on algebro-geometric…

This article is a review of theoretical advances in the research field of algebraic geometry and Bayesian statistics in the last two decades. Many statistical models and learning machines which contain hierarchical structures or latent…

This paper introduces arithmetic geometry for polynomial identity algebras using non-commutative (formal) deformation theory. Since formal deformation theory is inherently local the arithmetic and geometric results that follow give local…

We review some recent applications of machine learning to algebraic geometry and physics. Since problems in algebraic geometry can typically be reformulated as mappings between tensors, this makes them particularly amenable to supervised…

The main goal of the paper is the discussion of a deeper interaction between matrix theory over polynomial rings over a field and typical methods of commutative algebra and related algebraic geometry. This is intended in the sense of…

Consider the set of solutions to a system of polynomial equations in many variables. An algebraic manifold is an open submanifold of such a set. We introduce a new method for computing integrals and sampling from distributions on algebraic…

We aim at studying collections of algebraic structures defined over a commutative ring and investigating the complexity of significant constructions carried out on these objects. The assignment of measures of size, via a multiplicity…

We develop a collection of numerical algorithms which connect ideas from polyhedral geometry and algebraic geometry. The first algorithm we develop functions as a numerical oracle for the Newton polytope of a hypersurface and is based on…

Symmetry plays a central role in accelerating symbolic computation involving polynomials. This chapter surveys recent developments and foundational methods that leverage the inherent symmetries of polynomial systems to reduce complexity,…

We seek to determine a real algebraic variety from a fixed finite subset of points. Existing methods are studied and new methods are developed. Our focus lies on aspects of topology and algebraic geometry, such as dimension and defining…