Related papers: A Quaternionic Nullstellensatz

The classical Hilbert specialization property is a field-theoretic tool ensuring that polynomial irreducibility over a field is preserved under specialization of some of the variables. We develop an integral counterpart by introducing the…

We use tools of additive combinatorics for the study of subvarieties defined by {\it high rank} families of polynomials in high dimensional $\mathbb{F} _q$-vector spaces. In the first, analytic part of the paper we prove a number properties…

In this paper we prove that, for any $n\in \mathbb N$, the ideal generated by $n$ slice regular functions $f_1,\ldots,f_n$ having no common zeros concides with the entire ring of slice regular functions. The proof required the study of the…

We prove a negative solution to the analogue of Hilbert's tenth problem for rings of one variable non-Archimedean entire functions in any characteristic. In the positive characteristic case we prove more: the ring of rational integers is…

We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and $q$-difference equations for these polynomials. A general functional equation is found which allows one to relate…

Several sets of quaternionic functions are described and studied with respect to hy-perholomorphy, addition and (non commutative) multiplication, on open sets of H, then Hamil-ton 4-manifolds analogous to Riemann surfaces, for H instead of…

Let $K$ be an {\em arbitrary} field of characteristic $p>0$ and $\CD (P_n)$ be the ring of differential operators on a polynomial algebra $P_n$ in $n$ variables. A long anticipated {\em analogue of the inequality of Bernstein} is proved for…

We prove a factorization formula for the Taylor series coefficients of a zero of a polynomial as a function of the polynomial's coefficients. This result extends to more general functions which we call "complex-exponent polynomials". To…

We give a short proof of the most general version of the Nullstellensatz without using the Axiom of Choice.

This paper considers the extension of classical Lagrange interpolation in one real or complex variable to "polynomials of one quaternionic variable". To do this we develop some aspects of the theory of such polynomials. We then give a…

This paper studies the singularities of Cullen-regular functions of one quaternionic variable. The quaternionic Laurent series prove to be Cullen-regular. The singularities of Cullen-regular functions are thus classified as removable,…

In this paper, we introduce the quaternionic slice polyanalytic functions and we prove some of their properties. Then, we apply the obtained results to begin the study of the quaternionic Fock and Bergman spaces in this new setting. In…

We describe the class of n-variable polynomial functions that satisfy Acz\'el's bisymmetry property over an arbitrary integral domain of characteristic zero with identity.

We investigate the least studied class of differential rings -- the class of differential rings of nonzero characteristic. We present the notion of differentially closed quasifield and develop geometrical theory of differential equations in…

A long-standing open problem in representation stability is whether every finitely generated commutative algebra in the category of strict polynomial functors satisfies the noetherian property. In this paper, we resolve this problem…

We present a generalization of the Jacobian Conjecture for m polynomials in n variables: f1,...,fm belonging to k[x1,...,xn], where k is a field of characteristic zero and m=1,...,n. We express the generalized Jacobian condition in terms of…

We prove a four dimensional version of the Bernstein Theorem, with complex polynomials being replaced by quaternionic polynomials. We deduce from the theorem a quaternionic Bernstein's inequality and give a formulation of this last result…

Let $D$ be a positive definite quaternion algebra over a totally real number field $K$, $F(X,Y)$ a hermitian form in 2N variables over $D$, and $Z$ a right $D$-vector space which is isotropic with respect to $F$. We prove the existence of a…

We study generalizations of the classical Bernstein operators on polynomial spaces, where instead of fixing $\mathbf{1}$ and $x$, we require that $\mathbf{1}$ and a strictly increasing polynomial $f_1$ be fixed. Via several examples, we…

Within the framework of mappings between affine spaces, the notion of $n$-th polarization of a function will lead to an intrinsic characterization of polynomial functions. We prove that the characteristic features of derivations, such as…