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Given an ordinary differential field $K$ of characteristic zero, it is known that if $y$ and $1/y$ satisfy linear differential equations with coefficients in $K$, then $y'/y$ is algebraic over $K$. We present a new short proof of this fact…

We consider the problem of defining polynomials over function fields of positive characteristic. Among other results, we show that the following assertions are true. 1. Let $\G_p$ be an algebraic extension of a field of $p$ elements and…

We present a new approach to solving polynomial ordinary differential equations by transforming them to linear functional equations and then solving the linear functional equations. We will focus most of our attention upon the first-order…

Given a linear differential equation with coefficients in $\mathbb{Q}(x)$, an important question is to know whether its full space of solutions consists of algebraic functions, or at least if one of its specific solutions is algebraic.…

The solutions of the equation $f^{(p-1)} + f^p = h^p$ in the unknown function $f $over an algebraic function field of characteristic $p$ are very closely linked to the structure and factorisations of linear differential operators with…

Let $K$ be a field complete with respect to a discrete valuation $v$ of residue characteristic $p$. Let $f(z) \in K[z]$ be a separable polynomial of the form $z^\ell-c.$ Given $a \in K$, we examine the Galois groups and ramification groups…

A general method of obtaining linear differential equations having polynomial solutions is proposed. The method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some Lie algebra in the…

The existence of sufficiently many finite order meromorphic solutions of a differential equation, or difference equation, or differential-difference equation, appears to be a good indicator of integrability. In this paper, we investigate…

We present an algorithm for factoring linear differential operators with coefficients in a finite separable extension of F p (x). Our methods rely on specific tools arising in positive characteristic: p-curvature, structure of simple…

In [J. Lond. Math. Soc. 109 (2024), e12884, 22 pages, arXiv:2208.09721], the difference qKZ equations were considered modulo a prime number $p$ and a family of polynomial solutions of the qKZ equations modulo $p$ was constructed by an…

We consider the problem of solvability of linear differential equations over a differential field~$K$. We introduce a class of special differential field extensions, which widely generalizes the classical class of extensions of differential…

In this article, we study the growth of solutions of the homogeneous complex linear differential equation \begin{equation*} f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f^{\prime}+ A_{0}(z)f=0, \end{equation*}% where the coefficients…

In this paper, we prove that every iterative differential embedding problem over an algebraic function field in positive characteristic with an algebraically closed field of constants has a proper solution.

Let $K$ be the function field of a smooth projective geometrically integral curve over a finite extension of $\mathbb{Q}_p$. Following the works of Harari, Scheiderer, Szamuely, Izquierdo, and Tian, we study the local-global and weak…

The Jacobian conjecture over a field of characteristic zero is considered directly in view of the nonlinear partial differential equations it is associated with. Exploring the integrals of such partial differential equations, this work…

We formulate a conjecture classifying algebraic solutions to (possibly non-linear) algebraic differential equations, in terms of the primes appearing in the denominators of the coefficients of their Taylor expansion at a non-singular point.…

Let $k$ be a finite field, and $L$ be a $q$-linearized polynomial defined over $k$ of $q$-degree $r$ ($L=\sum^r_{i=0}a_iZ^{q^i}$, with $a_i\in k$). This paper provides an algorithm to compute a characteristic polynomial of $L$ over a large…

We study the existence of Fuchsian differential equations in positive characteristic with nilpotent p-curvature, and given local invariants. In the case of differential equations with logarithmic local mononodromy, we determine the minimal…

The paper generalizes Lazarus Fuchs' theorem on the solutions of complex ordinary linear differential equations with regular singularities to the case of ground fields of arbitrary characteristic, giving a precise description of the shape…

In this paper, we have characterized the nature and form of solutions of the following non-linear delay-differential equation: $$f^{n}(z)+\sum_{i=1}^{n-1}b_{i}f^{i}(z)+q(z)e^{Q(z)}L(z,f)=P(z),$$ where $b_i\in\mathbb{C}$, $L(z,f)$ be a…