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]].