Related papers: Systems of equations with a single solution
We study sets of univariate hyperbolic polynomials that share the same first few coefficients and show that they have a natural combinatorial description akin to that of polytopes. We define a stratification of such sets in terms of root…
A simple method to obtain a canonical partition function for one dimensional lattice gas model is presented. The simplification is based upon rewriting a sum over all possible configurations to a sum over numbers of clusters in the system.
Lattice polytope representation of natural numbers is introduced based on the fundamental theorem of arithmetic. The combinatorial and geometric properties of the polytopes are studied using Polymake and Qhull software. The volume of the…
Generalized matrix Lotka-Volterra lattice equations are obtained in a systematic way from a "master equation" possessing a bicomplex formulation.
A method is presented that reduces the number of terms of systems of linear equations (algebraic, ordinary and partial differential equations). As a byproduct these systems have a tendency to become partially decoupled and are more likely…
We give explicit formulas as well as a quadratic time algorithm to solve (so called) generalized Vandermonde's systems of p linear equations and n variables. It allows in particular to find all (so called Lagrange's) interpolation polynoms…
In this work we consider a simple, approximate, tending toward exact, solution of the system of two usual Lotka-Volterra differential equations. Given solution is obtained by an iterative method. In any finite approximation order of this…
We give necessary and sufficient existence criteria, and methods for finding, continuous solutions of linear equations whose coefficients are polynomials.
We propose a method to compute the numerical solutions of a polynomial system in complete intersection. This algorithm makes use of Bezout matrices and need only linear algebra computations. All the calculations can be done in floating…
If $P$ is a lattice polytope (that is, the convex hull of a finite set of lattice points in $\mathbf{R}^n$), then every sum of $h$ lattice points in $P$ is a lattice point in the $h$-fold sumset $hP$. However, a lattice point in the…
Let E be a compact set of C of positive logarithmic capacity. Let us suppose that for every polynomial $P\not=id$ we have $P^{-1}(E)\not=E$. Then for all no constant polynomials f and g such that $f^{-1}(E)=g^{-1}(E)$ we have f=g.
We introduce two classes of discrete polynomials and construct discrete equations admitting a Lax representation in terms of these polynomials. Also we give an approach which allows to construct lattice integrable hierarchies in its…
The analysis of solutions to algebraic equations is further simplified. A couple of functions and their analytic continuation or root findings are required.
Inspired by work done for systems of polynomial exponential equations, we study systems of equations involving the modular $j$ function. We show general cases in which these systems have solutions, and then we look at certain situations in…
This paper presents a simplified method of expressing the solution to cubic equations in terms of function evaluation only. The method eliminates the need to manipulate the original coefficients of the cubic polynomial and makes the…
To compute solutions of sparse polynomial systems efficiently we have to exploit the structure of their Newton polytopes. While the application of polyhedral methods naturally excludes solutions with zero components, an irreducible…
We study orbit-finite systems of linear equations, in the setting of sets with atoms. Our principal contribution is a decision procedure for solvability of such systems. The procedure works for every field (and even commutative ring) under…
Numerical homotopy continuation methods for three classes of polynomial systems are presented. For a generic instance of the class, every path leads to a solution and the homotopy is optimal. The counting of the roots mirrors the resolution…
In this paper we presents further developments regarding the enrichment of the basic Theory of Order Completion. In particular, spaces of generalized functions are constructed that contain generalized solutions to all systems of continuous,…
When a problem has more than one solution, it is often important, depending on the underlying context, to enumerate (i.e., to list) them all. Even when the enumeration can be done in polynomial delay, that is, spending no more than…