Related papers: Complete polynomial vector fields on $\C^2$, Part …
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.
We first generalize a classical iteration formula for one variable holomorphic mappings to a formula for higher dimensional holomorphic mappings. Then, as an application, we give a short and intuitive proof of a classical theorem, due to H.…
We classify the pairs of polynomials $f,g$ over a field $K$, such that $f(X)-g(Y)$ has a factor of total degree at most 2. This was done by Y. Bilu for characteristic 0 fields $K$. As his method does not work in positive characteristic, we…
We construct the full linearisation functor which takes a graded bundle of degree $k$ (a particular kind of graded manifold) and produces a $k$-fold vector bundle. We fully characterise the image of the full linearisation functor and show…
In this paper, we characterize arbitrary polynomial vector fields on $S^n$. We establish a necessary and sufficient condition for a degree one vector field on the odd-dimensional sphere $S^{2n-1}$ to be Hamiltonian. Additionally, we…
We investigate the integrability of polynomial vector fields through the lens of duality in parameter spaces. We examine formal power series solutions annihilated by differential operators and explore the properties of the integrability…
We study the Hamiltonian vector field $v=(-\partial f/\partial w,\partial f/\partial z)$ on $\mathbb C^2$, where $f=f(z,w)$ is a polynomial in two complex variables, which is non-degenerate with respect to its Newton's polygon. We introduce…
We study a graded vector space of polynomials associated to a square matrix, defined by a finite difference condition along the rows. We show this space coincides with one defined by directional derivatives, and prove it is…
In physics, it is believed that the consistency of two dimensional conformal field theory follows from the bootstrap equation. In this paper, we introduce the notion of a full vertex algebra by analyzing the bootstrap equation, which is a…
We develop a "metrically selfdual" variational calculus for $c$-monotone vector fields between general manifolds $X$ and $Y$, where $c$ is a coupling on $X\times Y$. Remarkably, many of the key properties of classical monotone operators…
We investigate a family of permutation polynomials of finite fields of characteristic 2. Through a connection between permutation polynomials and quadratic forms, a general treatment is presented to characterize these permutation…
We study multivariate polynomials over `structured' grids. We begin by proposing an interpretation as to what it means for a finite subset of a field to be structured; we do so by means of a numerical parameter, the nullity. We then extend…
We describe the Segal $K$-theory of the symmetric monoidal category of finite-dimensional vector spaces over a perfect field $\mathbb{F}$ together with an automorphism, or, equivalently, the group-completion of the $E_\infty$-algebra of…
We study invariants and structures of Poisson fields of rational functions in two variables. For four particular families, we classify the members, establish criteria for isomorphisms and, with the exception of the Weyl Poisson field,…
The aim of this paper is twofold. Firstly, we give easy-to-handle criteria to determine whether a given family of subsets of a vector space is a neighbourhood basis of the origin for a complete vector topology. Then, we apply these criteria…
In this paper we introduce geometric tools to study the families of rational vector fields of a given degree over $\mathbb C\mathbb P^1$. To a generic vector field of such a parametric family we associate several geometric objects: a…
We give an overview of invariants of algebraic singularities over perfect fields. We then show how they lead to a synthetic proof of embedded resolution of singularities of 2-dimensional schemes.
We prove that the Kuznetsov--Polishchuk exceptional collections on rational homogeneous spaces of the symplectic groups $\mathrm{Sp}(2n,\mathbb{C})$ are full and consist of vector bundles. To achieve this, we construct several classes of…
In our previous paper entitled "Axiomatic differential geometry -towards model categories of differential geometry-, we have given a category-theoretic framework of differential geometry. As the first part of our series of papers concerned…
In this paper we construct a combinatorial algorithm of resolution of singularities for binomial ideals, over a field of arbitrary characteristic. This algorithm is applied to any binomial ideal. This means ideals generated by binomial…