Related papers: Surjective derivations in small dimensions

We observe algebraic derivations on an affine domain B defined over an algebraically closed field of characteristic 0, which are called locally finite derivations in commutative and non-commutative contexts in other references. We observe…

We prove that an irreducible polynomial derivation in positive characteristic is a Jacobian derivation if and only if there exists an n-1-element p-basis of its ring of constants. In the case of two variables we characterize these…

Let $R$ be an affine domain of characteristic zero with finite quotients. We prove that a polynomial map over $R$ is surjective if and only if it is surjective over $\hat{R_{\mathfrak{m}}}$, the completion of $R$ with respect to…

In this paper we show that the image of any locally finite $k$-derivation of the polynomial algebra $k[x, y]$ in two variables over a field $k$ of characteristic zero is a Mathieu subspace. We also show that the two-dimensional Jacobian…

Let k be an algebraically closed field of characteristic zero and let B be a finitely generated k-domain. We study semisimple derivations on B, with special emphasis on those whose eigenvalues are integers. For such derivations, after…

Given $p$ polynomials with coefficients in a commutative unitary integral ring $\mathcal{C}$ containing $\mathbb{Q}$, we define the notion of a generic Bernstein-Sato polynomial on an irreducible affine scheme $V \subset…

In this paper we investigate to what extent the results of Z. Wang and D. Daigle on nice derivations of the polynomial ring in three variables over a field k of characteristic zero extend to the polynomial ring over a PID R, containing the…

\c{S}tef\u{a}nescu proved an elegant factorization result for polynomials over discrete valuation domains [CASC'2014, Lecture Notes in Computer Science, Ed. by V. Gerdt, W. Koepf, W. Mayr, and E. Vorozhtsov, Springer, Berlin, {Vol.…

We prove that $\delta$-derivations of a simple finite-dimensional Lie algebra over a field of characteristic zero, with values in a finite-dimensional module, are either inner derivations, or, in the case of adjoint module, multiplications…

We will prove that the Pierce-Birkhoff Conjecture holds for non-singular two-dimensional affine real algebraic varieties over real closed fields, i.e., if W is such a variety, then every piecewise polynomial function on W can be written as…

In this paper, we study higher derivations of Jacobian type in positive characteristic. We give a necessary and sufficient condition for $(n-1)$-tuples of polynomials to be extendable in the polynomial ring in $n$ variables over an integral…

We prove the following conjecture by S. Carpentier, A. De Sole, and V. G. Kac: Let K be a differential field and R be a differential subring of K. Let M be a matrix whose elements are differential operators with coefficents in R. Then, if M…

Let $A$ be the polynomial ring over $k$ (a field of characteristic zero) in $n+1$ variables. The commuting derivations conjecture states that $n$ commuting locally nilpotent derivations on $A$, linearly independent over $A$, must satisfy…

We prove constructively the existence of surjective morphisms from affine space onto certain open subvarieties of affine space of the same dimension. For any algebraic set $Z\subset \mathbb{A}^{n-2}\subset \mathbb{A}^{n}$, we construct an…

In this paper, two structural results concerning low degree polynomials over finite fields are given. The first states that over any finite field $\mathbb{F}$, for any polynomial $f$ on $n$ variables with degree $d \le \log(n)/10$, there…

A standard theorem in nonsmooth analysis states that a piecewise affine function $F:\mathbb R^n\rightarrow\mathbb R^n$ is surjective if it is coherently oriented in that the linear parts of its selection functions all have the same nonzero…

Cilleruelo conjectured that for an irreducible polynomial $f \in \mathbb{Z}[X]$ of degree $d \geq 2$ one has $$\log\left[\mathrm{lcm}(f(1),f(2),\ldots f(N))\right]\sim(d-1)N\log N$$ as $N \to \infty$. He proved it in the case $d=2$ but it…

We prove that a $C^{\infty}$ semialgebraic local diffeomorphism of $\mathbb{R}^n$ with non-properness set having codimension greater than or equal to $2$ is a global diffeomorphism if $n-1$ suitable linear partial differential operators are…

This paper is a step in our program for proving the Piece-Birkhoff Conjecture for regular rings of any dimension (this would contain, in particular, the classical Pierce-Birkhoff conjecture which deals with polynomial rings over a real…

In this article, some factorization properties of polynomials over discrete valuation domains are elucidated. These properties along with the notion of Newton index of a polynomial leads to a generalization of the main result proved by…