We prove that if a homogeneous, continuously differentiable vector field is asymptotically stable, then it admits a Lyapunov function which is the ratio of two polynomials (i.e., a rational function). We further show that when the vector field is polynomial, the Lyapunov inequalities on both the rational function and its derivative have sums of squares certificates and hence such a Lyapunov fun...

In this paper, we consider approximation algorithms for optimizing a generic multivariate polynomial function in discrete (typically binary) variables. Such models have natural applications in graph theory, neural networks, error-correcting codes, among many others. In particular, we focus on three types of optimization models: (1) maximizing a homogeneous polynomial function in binary variable...

An algebraic branching program (ABP) is a directed acyclic graph, with a start vertex s, and end vertex t and each edge having a weight which is an affine form in F[x1, x2, . . . , xn]. An ABP computes a polynomial in a natural way, as the sum of weights of all paths from s to t, where the weight of a path is the product of the weights of the edges in the path. An ABP is said to be homogeneous ...

We obtain an explicit polynomial whose simple positive real roots provide the limit cycles which bifurcate from the periodic orbits of a family of cubic polynomial differential centers when it is perturbed inside the class of all cubic polynomial differential systems. The family considered is the unique family of weight–homogeneous polynomial differential systems of weight–degree 2 with a center.

Let X be a homogeneous polynomial vector field of degree 2 on S. We show that if X has at least a non–hyperbolic singularity, then it has no limit cycles. We give necessary and sufficient conditions for determining if a singularity of X on S is a center and we characterize the global phase portrait of X modulo limit cycles. We also study the Hopf bifurcation of X and we reduce the 16 Hilbert’s ...

We give a new purely algebraic proof of the Baker-Campbell-Hausdorff theorem, which states that homogeneous components formal expansion $\log(e^Ae^B)$ are Lie polynomials. Our is based on recurrence formula for these and lemma if under certain conditions commutator non-commuting variable given polynomial polynomial, then itself polynomial.

The Bertrand-Darboux integrability condition for a certain class of perturbed harmonic oscillators is studied from the viewpoint of the Birkhoff-Gustavson(BG)-normalization: By solving an inverse problem of the BG-normalization on computer algebra, it is shown that if the perturbed harmonic oscillator with a homogeneous cubic-polynomial potential and the perturbed harmonic oscillator with a hom...

We investigate non-homogeneous linear differential equations of the form x′′(t) + x′(t) − x (t) = p (t) where p (t) is either a polynomial or a factorial polynomial in t. We express the solution of these differential equations in terms of the coefficients of p (t), in the initial conditions, and in the solution of the corresponding homogeneous differential equation y′′(t) + y′(t) − y (t) = 0 wi...

The theory of apolarity was first developed by Clebsch, Lasker, Richw x mond, Sylvester, and Wakeford 10, 17, 20 . They were first interested in studying homogeneous polynomials of degree p and in q variables, and in expressing them as sums of pth powers of linear terms. The problem is to minimize the number of pth powers which are required in such a sum. For instance, a result due to Sylvester...