Related papers: On a computer-aided approach to the computation of…
We suggest an algorithm for derivation of the Picard--Fuchs system of Pfaffian equations for Abelian integrals corresponding to semiquasihomogeneous Hamiltonians. It is based on an effective decomposition of polynomial forms in the…
Polynomial systems over the binary field have important applications, especially in symmetric and asymmetric cryptanalysis, multivariate-based post-quantum cryptography, coding theory, and computer algebra. In this work, we study the…
New criteria are established for upper bounds on the number of limit cycles of periodic Abel differential equations having two periodic invariant curves, one of them bounded. The criteria are applied to obtain upper bounds of either zero or…
We first prove a one-to-one correspondence between finding Hamiltonian cycles in a cubic planar graphs and finding trees with specific properties in dual graphs. Using this information, we construct an exact algorithm for finding…
The hamiltonian circuit polytope is the convex hull of feasible solutions for the circuit constraint, which provides a succinct formulation of the traveling salesman and other sequencing problems. We study the polytope by establishing its…
Cylindrical algebraic decomposition (CAD) is an important tool for the study of real algebraic geometry with many applications both within mathematics and elsewhere. It is known to have doubly exponential complexity in the number of…
In this paper, we study the number of limit cycles which bifurcate from the periodic orbits of cubic polynomial vector fields of Lotka-Volterra type having a rational first integral of degree 2, under polynomial perturbations of degree $n$.…
We present a method for the calculation of electronic structure of systems that contain tens of thousands of atoms. The method is based on the division of the system into mutually overlapping fragments and the representation of the…
In this paper, we propose an incremental algorithm for computing cylindrical algebraic decompositions. The algorithm consists of two parts: computing a complex cylindrical tree and refining this complex tree into a cylindrical tree in real…
An analytical method for investigation of the evolution of dynamical systems {\it with independent on time accuracy} is developed for perturbed Hamiltonian systems. The error-free estimation using of computer algebra enables the application…
The representation of polynomials by arithmetic circuits evaluating them is an alternative data structure which allowed considerable progress in polynomial equation solving in the last fifteen years. We present a circuit based computation…
We describe several improvements to algorithms for the rigorous computation of the endomorphism ring of the Jacobian of a curve defined over a number field.
Approximating periodic solutions to the coupled Duffing equations amounts to solving a system of polynomial equations. The number of complex solutions measures the algebraic complexity of this approximation problem. Using the theory of…
Before we proposed an algebraic technics for the Hamiltonian approach to the evolution systems of partial differential equations, including systems with constraints. Here we further develop this approach and present the defining system of…
Often in the study the periodic orbits in dynamical systems, the computation of the Lyapunov Coeficients is needed. In this paper, the calculations of this coeficients were done via complex variable transformation in order to obtain the…
In this article we introduce algorithms which compute iterations of Gauss-Manin connections, Picard-Fuchs equations of Abelian integrals and mixed Hodge structure of affine varieties of dimension $n$ in terms of differential forms. In the…
Generalizing a method of Sutherland and the author for elliptic curves, we design a subexponential algorithm for computing the endomorphism rings of ordinary abelian varieties of dimension two over finite fields. Although its correctness…
The systematic method for the conversion of first class constraints to the equivalent set of Abelian one based on the Dirac equivalence transformation is developed. The representation for the corresponding matrix performing this…
We present algorithms for computation and visualization of amoebas, their contours, compactified amoebas and sections of three-dimensional amoebas by two-dimensional planes. We also provide method and an algorithm for the computation…
Let $p$ be an odd prime number. We propose an algorithm for computing rational representations of isogenies between Jacobians of hyperelliptic curves via-adic differential equations with a sharp analysis of the loss of precision.…