Related papers: Redundant Picard-Fuchs system for Abelian integral…
We consider an integrable polynomial system with generalized Darboux first integral H_0. We assume that it defines a family of real cycles in a region bounded by a polycycle. To any polynomial form \eta one can associate the pseudo-abelian…
We consider polynomials of degree $d$ with only real roots and a fixed value of discriminant, and study the problem of minimizing the absolute value of polynomials at a fixed point off the real line. There are two explicit families of…
We discuss the existence of an absolute Chow-Kuenneth decomposition for complete degenerations of families of Abelian threefolds with complex multiplication over a particular Picard Modular Surface studied by Holzapfel. In addition to the…
This is the first part of a two-part paper describing a new concept of separation of variables applied to the Clebsch integrable case of the Kirchhoff equations. There are two principal novelties: 1) Separating coordinates are constructed…
In this paper we give a conditional improvement to the Elekes-Szab\'{o} problem over the rationals, assuming the Uniformity Conjecture. Our main result states that for $F\in \mathbb{Q}[x,y,z]$ belonging to a particular family of…
We give a uniform asymptotic bound for the number of zeros of complete Abelian integrals in domains bounded away from infinity and the singularities.
We answer the following long-standing question of Kolchin: given a system of algebraic-differential equations $\Sigma(x_1,\dots,x_n)=0$ in $m$ derivatives over a differential field of characteristic zero, is there a computable bound, that…
{We explore a simple {\it geometric model} for functions between spaces of the same dimension (in infinite dimensions, we require that Jacobians be Fredholm operators of index zero). The model combines standard results in analysis and…
We obtain nontrivial solutions of a critical fractional $p$-Laplacian equation in the whole space and with possibly vanishing potentials. In addition to the usual difficulty of the lack of compactness associated with problems involving…
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…
The present work is the first of a serie of two papers, in which we analyse the higher variational equations associated to natural Hamiltonian systems, in their attempt to give Galois obstruction to their integrability. We show that the…
We provide an upper bound for the number of limit cycles that planar polynomial differential systems of a given degree may have. The bound turns out to be a polynomial of degree four in the degree of the system. The strategy brings together…
In this paper, we approach the problem of finding the zeros of the sum of a maximally monotone operator and a monotone and Lipschitz continuous one in a real Hilbert space via an implicit forward-backward-forward dynamical system with…
We present some new and recent algorithmic results concerning polynomial system solving over various rings. In particular, we present some of the best recent bounds on: (a) the complexity of calculating the complex dimension of an algebraic…
In this paper, we study the minimal number of elements of maximal order within a zero-sumfree sequence in a finite Abelian p-group. For this purpose, in the general context of finite Abelian groups, we introduce a new number, for which…
In the paper we give an upper estimate of the number of apparent singularities that are sufficient for construction of a system of linear differential equations on a Riemann surface with given fuchsian singularities and monodromy.
We study second-order modular differential equations whose solutions transform equivariantly under the modular group. In the reducible case, we construct all such solutions using an explicit ansatz involving Eisenstein series and the…
This paper describes a quantum algorithm for efficiently decomposing finite Abelian groups. Such a decomposition is needed in order to apply the Abelian hidden subgroup algorithm. Such a decomposition (assuming the Generalized Riemann…
We introduce a new numerical method, based on Bernoulli polynomials, for solving multiterm variable-order fractional differential equations. The variable-order fractional derivative was considered in the Caputo sense, while the…
We generalize the classical lifting and recombination scheme for rational and absolute factorization of bivariate polynomials to the case of a critical fiber. We explore different strategies for recombinations of the analytic factors,…