Given a 3-dimensional vector eld V with coordinates V x , V y and V z that are homogeneous polynomials in the ring kx; y; z], we give a necessary and suucient condition for the existence of a Liouvillian rst integral of V , which is homogeneous of degree 0. This condition is the existence of some 1-forms with coordinates in ring kx; y; z] enjoying precise properties; in particular, they have to...

Observational data hint at a finite universe, with spherical manifolds such as the Poincaré dodecahedral space tentatively providing the best fit. Simulating the physics of a model universe requires knowing the eigenmodes of the Laplace operator on the space. The present article provides explicit polynomial eigenmodes for all globally homogeneous 3-manifolds: the Poincaré dodecahedral space S3/...

We provide the 22 different global phase portraits in the Poincaré disk of all centers of the so called Kukles polynomial differential systems of the form ẋ = −y, ẏ = x + Q5(x, y), where Q5 is a real homogeneous polynomial of degree 5 defined in R.

This paper suggests a detailed algorithm for computation of the Jacobson form of the polynomial matrix associated with the transfer matrix describing the multi-input multi-output nonlinear control system, defined on homogeneous time scale. The algorithm relies on the theory of skew polynomial rings.

We present a new algorithm for the computation of resultants associated with multihomogeneous (and, in particular, homogeneous) polynomial equation systems using straight-line programs. Its complexity is polynomial in the number of coefficients of the input system and the degree of the resultant computed.

The multigraded Hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by an abelian group with a fixed Hilbert function. We prove that any multigraded Hilbert scheme is smooth and irreducible when the polynomial ring is Z[x, y], which establishes a conjecture of Haiman and Sturmfels. © 2009 Gregory G. Smith. Published by Elsevier Inc. All rights reserved.

Let f (x,y,z) is a cyclic homogeneous polynomial of degree four of three nonnegative real variables satisfying the condition f (1,1,1) = 0 . We find necessary and sufficient condition to be true the inequality f (x,y,z) 0 , for this aim we introduce a characteristic polynomial Jf (t) and by its root t0 > 0 we formulate the condition.

Multi-homogeneous polynomial systems arise in many applications. We provide bit complexity estimates for solving them which, up to a few extra other factors, are quadratic in the number of solutions and linear in the height of the input system under some genericity assumptions. The assumptions essentially imply that the Jacobian matrix of the system under study has maximal rank at the solution ...

We classify all the centers of a planar weight–homogeneous polynomial vector field of weight degree 1, 2, 3 and 4.