Related papers: Existence and convergence of Puiseux series soluti…

In this paper, we study the algebraic, rational and formal Puiseux series solutions of certain type of systems of autonomous ordinary differential equations. More precisely, we deal with systems which associated algebraic set is of…

There exist several methods for computing exact solutions of algebraic differential equations. Most of the methods, however, do not ensure existence and uniqueness of the solutions and might fail after several steps, or are restricted to…

The existence, uniqueness and convergence of formal Puiseux series solutions of non-autonomous algebraic differential equations of first order at a nonsingular point of the equation is studied, including the case where the celebrated…

In this paper we study systems of autonomous algebraic ODEs in several differential indeterminates. We develop a notion of algebraic dimension of such systems by considering them as algebraic systems. Afterwards we apply differential…

I begin from a particular field of generalised Puiseux series and investigate a class of nonlinear differential equations in the field. It is appeared that the main part of differential equation determines solvability and positions of…

We study differential equations $F(y,...,y^{(n)})=0$ where $F(Y_0,...,Y_n)$ is a formal series in $Y_0,...,Y_n$ with coefficients in some field of \emph{generalized power series} $\mathds{K}_r$ with finite rank $r\in\mathbb{N}^*$. Our…

This final degree project is devoted to study the topological classification of complex plane curves. These are subsets of $\mathbb{C}^2$ that can be described by an equation $f(x,y)=0$. Loosely speaking, curves are said to be equivalent in…

Let $F\in \mathbb{K}[X, Y ]$ be a polynomial of total degree $D$ defined over a perfect field $\mathbb{K}$ of characteristic zero or greater than $D$. Assuming $F$ separable with respect to $Y$ , we provide an algorithm that computes the…

For a second order linear differential equation $f''+A(z)f'+B(z)f=0$, with $ A(z)$ and $B(z)$ being transcendental entire functions under some restriction, we have established that all non-trivial solutions are of infinite order. In…

A method for finding Puiseux series goes back to Isaac Newton, which gives the terms of Puiseux series through an infinite recursive process; an additional argument is then used to show that the resulting Puiseux series are convergent. This…

The recent significant enrichment of the Order Completion Method for nonlinear Systems of PDEs resulted in the global existence of generalized solutions to a large class of such equations. In this paper we investigate the existence and…

We study higher order linear differential equation $y^{(k)}+A_1(z)y=0$ with $k\geq2$, where $A_1=A+h$, $A$ is a transcendental entire function of finite order with $\frac{1}{2}\leq \mu(A)<1$ and $h\neq0$ is an entire function with…

Using nonstandard methods, we show that the time dependent Fourier series of any smooth function F, solving the wave equation, on a finite closed interval, with vanishing boundary conditions, converges uniformly to F.

The problem of algebraic dependence of solutions to (non-linear) first order autonomous equations over an algebraically closed field of characteristic zero is given a `complete' answer, obtained independently of model theoretic results on…

We consider the problem of deciding whether a common solution to a multivariate polynomial equation system is isolated or not. We present conditions on a given truncated Puiseux series vector centered at the point ensuring that it is not…

Let $f(t,y,y')=\sum_{i=0}^n a_i(t,y)y'^i=0$ be an irreducible first order ordinary differential equation with polynomial coefficients. Eremenko in 1998 proved that there exists a constant $C$ such that every rational solution of…

We present in this paper a detailed note on the computation of Puiseux series solutions of the Riccatti equation associated with a homogeneous linear ordinary differential equation. This paper is a continuation of [1] which was on the…

In this paper we present a family of values of the parameters of the third Painlev\'{e} equation such that Puiseux series formally satisfying this equation -- considered as series of $z^{2/3}$ -- are series of exact Gevrey order one. We…

We discuss the solvability of an infinite system of first order ordinary differential equations on the half line, subject to nonlocal initial conditions. The main result states that if the nonlinearities possess a suitable "sub-linear"…

Consider the differential equation $y'=F(x,y)$. We determine the weakest possible upper bound on $|F(x,y)-F(x,z)|$ which guarantees that this equation has for all initial values a unique solution, which exists globally.