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The classical Liouvile integrability means that there exist $n$ independent first integrals in involution for $2n$-dimensional phase space. However, in the infinite-dimensional case, an infinite number of independent first integrals in…

Let D be a domain with quotient field K and A a D-algebra. We call a polynomial with coefficients in K that maps every element of A to an element of A "integer-valued on A". For commutative A we also consider integer-valued polynomials in…

In different areas of discrete mathematics, a certain type of polynomials, having coefficients in a field K of finite characteristic, has been considered. The form and the degree of these polynomials, here called projective, are simply…

The aim of this paper is to provide sufficient conditions for when a polynomial or rational function over a field K is prime using its order of vanishing at infinity and the resultant.

In 1992, P. Pol\'{a}\v{c}ik\cite{P2} showed that one could linearly imbed any vector fields into a scalar semi-linear parabolic equation on $\Omega$ with Neumann boundary condition provided that there exists a smooth vector field…

Some more general "inheritance conditions" have been found for a given set of symmetry generators $\{\mathbf{Z}_{\bar{l}}\}$ acting on some set of coupled ordinary differential equations, once the "first integration method" has been applied…

Let $F(t,u)\equiv F(u)$ be a formal power series in $t$ with polynomial coefficients in $u$. Let $F\_1, ..., F\_k$ be $k$ formal power series in $t$, independent of $u$. Assume all these series are characterized by a polynomial equation $$…

A polynomial with coefficients in the ring of integers $\mathcal{O}_{K}$ of a global field $K$ is called intersective if it has a root modulo every finite-indexed subgroup of $\mathcal{O}_{K}$. We prove two criteria for a polynomial…

We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some…

Let $V$ be a vector space over a field $k, P:V\to k, d\geq 3$. We show the existence of a function $C(r,d)$ such that $rank (P)\leq C(r,d)$ for any field $k,char (k)>d$, a finite-dimensional $k$-vector space $V$ and a polynomial $P:V\to k$…

We establish a Primitive Element Theorem for fields equipped with several commuting operators such that each of the operators is either a derivation or an automorphism. More precisely, we show that for every extension $F \subset E$ of such…

The First Fundamental Theorem of Invariant Theory describes a minimal generating set of the invariant polynomial ring under the action of some group $G$. In this note we give an elementary and direct proof for the $\operatorname{GL}_2(K)$…

We prove that any globally periodic rational discrete system in K^k(where K denotes either R or C), has unconfined singularities, zero algebraic entropy and it is complete integrable (that is, it has as many functionally independent first…

Our goal is to settle the following faded problem: The Jacobian Conjecture (JC_n): If f_1,..,f_n are elements in a polynomial ring k[X_1,..,X_n] over a field k of characteristic 0 such that det(\partial f_i/ \partial X_j) is a nonzero…

Let $K$ be a global field and $n > 1$ an integer. We show $n$ is composite if and only if there is an irreducible polynomial $f(x) \in K[x]$ of degree $n$ which is reducible $q$-adically for all the primes $q$ of $K$.

We characterize finite Galois extensions $K$ of the field of rational numbers in terms of the rings ${\rm Int}_{\mathbb{Q}}(\mathcal O_K)$, recently introduced by Loper and Werner, consisting of those polynomials which have coefficients in…

Using the Galois theory over function field, and the holomorphy of algebroids defined via irreducible polynomial at singular points, we prove the injectivity of any kellerian mapping. The famous Jacobian conjecture is true.

Polynomials and elements over finite fields exhibit closely related algebraic structures, and many properties defined for elements extend naturally to polynomials. The concepts of order and $\mathbb{F}_q$-Order for elements have been…

The Jacobian Conjecture states that any locally invertible polynomial system in C^n is globally invertible with polynomial inverse. C. W. Bass et al. (1982) proved a reduction theorem stating that the conjecture is true for any degree of…

We investigate the interplay between invariant varieties of vector fields and the inflection locus of linear systems with respect to the vector field. Among the consequences of such investigation we obtain a computational criteria for the…