Related papers: Classification of Complex Polynomial Vector Fields…
In this paper we address the following questions: (i) Let $C\subset \mathbb C^2$ be an orbit of a polynomial vector field which has finite total Gaussian curvature. Is $C$ contained in an algebraic curve? (ii) What can be said of a…
We construct a polynomial planar vector field of degree two with one invariant algebraic curves of large degree. We exhibit an explicit quadratic vector fields which invariant curves of degree nine, twelve, fifteen and eighteen degree.
We give the classification, up to homeomorphisms, of reduced complex polynomials with 2 variables with one critical value.
Several classes of systems of evolution equations with one or two vector unknowns are considered. We investigate also systems with one vector and one scalar unknown. For these classes all equations having the simplest higher symmetry are…
We give a first-order definition of key polynomials, we show the links with previous definitions, that it is relevant to study key degrees, and to use a kind of valuations that we call partially multiplicative. We also prove or reprove…
To a univariate monic polynomial is attached a special planar forest that is called the picture of the polynomial. Isotopy classes of pictures are called signatures. All combinatorially possible signatures are realized and spaces of…
In this paper we give a generalization of injective and projective complexes.
In this paper, we give an explicit description of holomorphic polyvector fields on smooth compact toric varieties, which generalizes Demazure's result of holomorphic vector fields on toric varieties.
In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed.
We characterize those valued fields for which the image of the valuation ring under every polynomial in several variables contains an element of maximal value, or zero.
Given a simplicial complex whose vertices are labeled with positive integers, one can associate a vector configuration whose corresponding toric variety is the Zariski closure of a hierarchical model. We classify all the vertex-weighted…
We present a list of all polynomial dominanting maps of the complex plane with branched value curve isomorphic to the complex line, up to polynomial automorphisms.
For any compact oriented manifold $M$, we show that that the top degree multi-vector fields transverse to the zero section of $\wedge^{\text{top}}TM$ are classified, up to orientation preserving diffeomorphism, in terms of the topology of…
In this paper, complex vector bundles of rank $r$ over $8$-dimensional spin$^{c}$ manifolds are classified in terms of the Chern classes of the complex vector bundles and the cohomology ring of the manifolds, where $r = 3$ or $4$. As an…
We define, for an arbitrary partially ordered set, a multi-variable polynomial generalizing the hook polynomial.
We study the homotopy theory of diagrams of chain complexes over a field indexed by a finite poset, and show that it can be completely described in terms of appropriate diagrams of graded vector spaces.
Indices of vector fields on (complex analytic) singular varieties have been considered by various authors from several different viewpoints. All these indices coincide with the classical local index of Poincar\'e-Hopf when the ambient…
We describe a conjectural classification of Poisson vertex algebras of CFT type and of Poisson vertex algebras in one differential variable (= scalar Hamiltonian operators).
We study the stratification of the space of monic polynomials with real coefficients according to the number and multiplicities of real zeros. In the first part, for each of these strata we provide a purely combinatorial chain complex…
Let K be a field. For a given valuation on K[x], we determine the structure of its graded algebra and describe its set of key polynomials, in terms of any given key polynomial of minimal degree. We also characterize valuations not admitting…