Related papers: On Abel's problem and Gauss congruences
Abel's problem consists in identifying the conditions under which the diferential equation $y'=\eta y$, with $\eta$ an algebraic function in $\mathbb{C}(x)$, possesses a non-zero algebraic solution $y$. This problem has been algorithmically…
We study nonconstant rational solutions of \[ x'=A_3(t)x^{n_3}+A_2(t)x^{n_2}+A_1(t)x^{n_1}, \qquad 1<n_1<n_2<n_3, \] with $A_i\in\Bbbk[t]$, $\Bbbk\in\{\mathbb R,\mathbb C\}$. We prove that every such solution is of the form $x=1/p(t)$, and…
Given an autonomous first order algebraic ordinary differential equation F(y,y')=0, we prove that every formal Puiseux series solution, expanded around any finite point or at infinity, is convergent. The proof is constructive and we provide…
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…
A constructive version of Newton-Puiseux theorem for computing the Puiseux expansions of algebraic curves is presented. The proof is based on a classical proof by Abhyankar. Algebraic numbers are evaluated dynamically; hence the base field…
We develop techniques at the interface between differential algebra and model theory to study the following problems of exponential algebraicity: Does a given algebraic differential equation admits an exponentially algebraic solution, that…
We give an exact characterization of admissibility in statistical decision problems in terms of Bayes optimality in a so-called nonstandard extension of the original decision problem, as introduced by Duanmu and Roy. Unlike the…
Linear second order differential equations having a large real parameter and turning point in the complex plane are considered. Classical asymptotic expansions for solutions involve the Airy function and its derivative, along with two…
This work contributes to the domains of Boolean algebra and of Bayesian probability, by proposing an algebraic extension of Boolean algebras, which implements an operator for the Bayesian conditional inference and is closed under this…
This paper is dedicated to present an exact solution for a nonlinear differential equation so-called Abel equation. This equation was known as one of the group of unsolvable differential equations. The present method is applicable for any…
We provide a new arithmetic characterization for the sequence of coefficients of algebraic power series $f(t)$ having the property that the differential equation $y'(t) = f(t) y(t)$ has algebraic solutions only. This extends a recent result…
We develop a unified analytical and computational framework for the generalized Abel ordinary differential equation $y^{\prime }(x)=a_n(x)\bigl(% y^n+\lambda_{n-1}(x)y^{n-1}+\dots+\lambda_0(x)\bigr)$ of arbitrary degree $% n\ge1$ on the…
Applying symmetry reduction to a class of $\mathrm{SL}(2,\mathbb R)$-invariant third-order ODEs, we obtain Abel equations whose general solution can be parametrised by hypergeometric functions. Particular case of this construction provides…
Given $\eta=\begin{pmatrix} a&b\\c&d \end{pmatrix}\in \text{GL}_2(\mathbb{Q})$, we consider the number of rational points on the genus one curve \[H_\eta:y^2=(a(1-x^2)+b(2x))^2+(c(1-x^2)+d(2x))^2.\] We prove that the set of $\eta$ for which…
Is it possible to distinguish algebraic from transcendental real numbers by considering the $b$-ary expansion in some base $b\ge2$? In 1950, \'E. Borel suggested that the answer is no and that for any real irrational algebraic number $x$…
An elementary proof is given for a nonterminating "strange" cubic $_7F_6$-series summation formula of Gasper and Rahman, through the modified Abel lemma on summation by parts. As a byproduct, an interesting nonterminating…
Solvable Lie algebras having at least one Abelian descending central ideal are studied. It is shown that all such Lie algebras can be built up from canonically defined ideals. The nature of such ideals is elucidated and their construction…
Chu has recently shown that the Abel lemma on summations by parts can serve as the underlying relation for Bailey's ${}_6\psi_6$ bilateral summation formula. In other words, the Abel lemma spells out the telescoping nature of the…
In this article, using only elementary knowledge of complex numbers, we sketch a proof of the celebrated Abel--Ruffini theorem, which states that the general solution to an algebraic equation of degree five or more cannot be written using…
We introduce tropical Newton-Puiseux polynomials admitting rational exponents. A resolution of a tropical hypersurface is defined by means of a tropical Newton-Puiseux polynomial. A polynomial complexity algorithm for resolubility of a…