Related papers: On global non-oscillation of linear ordinary diffe…

For a class of ordinary differential operators $P$ with polynomial coefficients, we give a necessary and sufficient condition for $P$ to be globally regular in $\R$, i.e. $u\in\cS^\prime(\R)$ and $Pu\in\cS(\R)$ imply $u\in \cS(\R)$ (this…

Differential systems with a Fuchsian linear part are studied in regions including all the singularities in the complex plane of these equations. Such systems are not necessarily analytically equivalent to their linear part (they are not…

We study the problem of placing effective upper bounds for the number of zeros of solutions of Fuchsian systems on the Riemann sphere. The principal result is an explicit (non-uniform) upper bound, polynomially growing on the frontier of…

We show that for every second order Fuchsian linear differential equation $E$ with $n$ singularities of which $n-3$ are apparent there exists a hypergeometric equation $H$ and a linear differential operator with polynomial coefficients…

We prove the classical result, which goes back at least to Fourier, that a polynomial with real coefficients has all zeros real and distinct if and only if the polynomial and also all of its nonconstant derivatives have only negative minima…

We address some global solvability issues for classes of smooth nonsingular vector fields $L$ in the plane related to cohomological equations $Lu=f$ in geometry and dynamical systems. The first main result is that $L$ is not surjective in…

We investigate the probability that a random polynomial with independent, mean-zero and finite variance coefficients has no real zeros. Specifically, we consider a random polynomial of degree $2n$ with coefficients given by an i.i.d.…

We contribute to the exceptional APN conjecture by showing that no polynomial of degree m = 2 r (2 {\ell} + 1) where gcd(r, {\ell}) 2, r 2, {\ell} 1 with a nonzero second leading coefficient can be APN over infinitely many extensions of the…

We consider deformations of $2\times2$ and $3\times3$ matrix linear ODEs with rational coefficients with respect to singular points of Fuchsian type which don't satisfy the well-known system of Schlesinger equations (or its natural…

This paper consists of three parts: First, letting $b_1(z)$, $b_2(z)$, $p_1(z)$ and $p_2(z)$ be nonzero polynomials such that $p_1(z)$ and $p_2(z)$ have the same degree $k\geq 1$ and distinct leading coefficients $1$ and $\alpha$,…

We give some sufficient conditions that ensure oscillations and nonoscillations for nonautonomous impulsive differential equations with piecewise constant arguments of generalized type. We cover several cases of differential equations with…

We look for spectral type differential equations satisfied by the generalized Jacobi polynomials, which are orthogonal on the interval [-1,1] with respect to a weight function consisting of the classical Jacobi weight function together with…

Let $\Phi_{\Lambda_{n}}$ be the unique solution of the differential operator $L=\prod_{j=0}^{n}\left( \frac{d}{dx}-\lambda_{j}\right) $ such that $\Phi_{\Lambda_{n}}^{\left( j\right) }\left( 0\right) =0$ for $j=0,...,n-1,$ and…

We consider parametrized systems of generalized polynomial equations (with real exponents) in $n$ positive variables, involving $m$ monomials with positive parameters; that is, $x\in\mathbb{R}^n_>$ such that ${A \, (c \circ x^B)=0}$ with…

We consider singular perturbations of eigenvalue problems. We prove that to these problems correspond simple eigenvalues and we study their asymptotic behavior. As a result, we prove global bifurcation results for non uniformly and fully…

Let $\mathcal F$ be a foliation on a smooth projective surface $S$ over the complex number $\mathbb{C}$. We introduce three birational non-negative invariants $c_1^2(\mathcal F)$, $c_2(\mathcal F)$ and $\chi(\mathcal F)$, called the Chern…

We show that Newton's method converges globally at a linear rate for objective functions whose Hessians are stable. This class of problems includes many functions which are not strongly convex, such as logistic regression. Our linear…

We consider deformations of a differential system with Poincare' rank 1 at infinity and Fuchsian singularity at zero along a stratum of a coalescence locus. We give necessary and sufficient conditions for the deformation to be strongly…

We show that if m is a probability measure with infinite support on the unit circle having no singular component and a differentiable weight, then the corresponding paraorthogonal polynomial P_n(z;B) solves an explicit second order linear…

In this paper, we provide a complete Plancherel-Rotach asymptotic analysis of polynomials that satisfy a second-order difference equation with linear coefficients. According to the signs of the parameters, we classify the difference…