Dans cet article, nous étudions les modules libres de type fini sur l’anneau $\mathscr{H}[d/dx]$ où $\mathscr{H}$ est l’anneau des éléments analytiques dans une couronne $r_1\<\left|x\right|\<r_2$ de ${ℂ}_p$. D’une part, nous définissons, pour chaque nombre $r$ de $[r_1,r_2]$, un rayon de convergence “générique" et nous montrons que celui-ci dépend continûment de $r$. D’autre part, nous étudions l’existence et l’unicité d’un “antécédent de Frobenius".

Let X be a Leibniz algebra with unit e, i.e. an algebra with a right invertible linear operator D satisfying the Leibniz condition: D(xy) = xDy + (Dx)y for x,y belonging to the domain of D. If logarithmic mappings exist in X, then cosine and sine elements C(x) and S(x) defined by means of antilogarithmic mappings satisfy the Trigonometric Identity, i.e. ${\left[C\left(x\right)\right]}^2+{\left[S\left(x\right)\right]}^2=e$ whenever x belongs to the domain of these mappings. The following question arises: Do there exist non-Leibniz algebras with logarithms such that...

In this article, we introduce a special class of non complete webs, the NN-webs. We also study the algebraic and geometric properties of these webs.

Let $P$ be a poset and $d$ be a derivation on $P$. In this research, the notion of generalized $d$-derivation on partially ordered sets is presented and studied. Several characterization theorems on generalized $d$-derivations are introduced. The properties of the fixed points based on the generalized $d$-derivations are examined. The properties of ideals and operations related with generalized $d$-derivations are studied.

Let k be a field. We prove that any polynomial ring over k is a Kadison algebra if and only if k is infinite. Moreover, we present some new examples of Kadison algebras and examples of algebras which are not Kadison algebras.

Let k be a field of chracteristic p > 0. We describe all derivations of the polynomial algebra k[x,y], homogeneous with respect to a given weight vector, in particular all monomial derivations, with the ring of constants of the form $k[x^p,y^p,f]$, where $f\in k[x,y]\setminus k[x^p,y^p]$.

Let d be a k-derivation of k[x,y], where k is a field of characteristic zero. Denote by $\tilde{d}$ the unique extension of d to k(x,y). We prove that if ker d ≠ k, then ker $\tilde{d}$ = (ker d)0, where (ker d)0 is the field of fractions of ker d.

We show that the GVC (generalized vanishing conjecture) holds for the differential operator $\Lambda =({\partial }_x-\Phi \left({\partial }_y\right)){\partial }_y$ and all polynomials $P(x,y)$, where $\Phi \left(t\right)$ is any polynomial over the base field. The GVC arose from the study of the Jacobian conjecture.

In this paper, we give a geometrization and a generalization of a lemma of differential Galois theory, used by Singer and van der Put in their reference book. This geometrization, in addition of giving a nice insight on this result, offers us the opportunity to investigate several points of differential algebra and differential algebraic geometry. We study the class of simple Δ-schemes and prove that they all have a coarse space of leaves. Furthermore, instead of considering schemes endowed with...

Let Δ denote the discriminant of the generic binary d-ic. We show that for d ≥ 3, the Jacobian ideal of Δ is perfect of height 2. Moreover we describe its SL2-equivariant minimal resolution and the associated differential equations satisfied by Δ. A similar result is proved for the resultant of two forms of orders d, e whenever d ≥ e-1. If Φn denotes the locus of binary forms with total root multiplicity ≥ d-n, then we show that the ideal of Φn is also perfect, and we construct a covariant which...

Let k[[x,y]] be the formal power series ring in two variables over a field k of characteristic zero and let d be a nonzero derivation of k[[x,y]]. We prove that if Ker(d) ≠ k then Ker(d) = Ker(δ), where δ is a jacobian derivation of k[[x,y]]. Moreover, Ker(d) is of the form k[[h]] for some h ∈ k[[x,y]].

We describe the ring of constants of a specific four variable Lotka-Volterra derivation. We investigate the existence of polynomial constants in the other cases of Lotka-Volterra derivations, also in n variables.

The notion of a closed polynomial over a field of zero characteristic was introduced by Nowicki and Nagata. In this paper we discuss possible ways to define an analog of this notion over fields of positive characteristic. We are mostly interested in conditions of maximality of the algebra generated by a polynomial in a respective family of rings. We also present a modification of the condition of integral closure and discuss a condition involving partial derivatives.