Derived functors of and abelian ab3*- and Ab4*-categories with enough injectives
On the formal inverse of polynomial endomorphisms
A counterexample to a conjecture of Bass, Connell and Wright
Let F=X-H: → be a polynomial map with H homogeneous of degree 3 and nilpotent Jacobian matrix J(H). Let G=(G1,...,Gn) be the formal inverse of F. Bass, Connell and Wright proved in [1] that the homogeneous component of of degree 2d+1 can be expressed as , where T varies over rooted trees with d vertices, α(T)=CardAut(T) and is a polynomial defined by (1) below. The Jacobian Conjecture states that, in our situation, is an automorphism or, equivalently, is zero for sufficiently large d....
Polynomial algebra of constants of the four variable Lotka-Volterra system
We describe the ring of constants of a specific four variable Lotka-Volterra derivation. We investigate the existence of polynomial constants in the other cases of Lotka-Volterra derivations, also in n variables.
Rings of constants of generic 4D Lotka-Volterra systems
We show that the rings of constants of generic four-variable Lotka-Volterra derivations are finitely generated polynomial rings. We explicitly determine these rings, and we give a description of all polynomial first integrals of their corresponding systems of differential equations. Besides, we characterize cofactors of Darboux polynomials of arbitrary four-variable Lotka-Volterra systems. These cofactors are linear forms with coefficients in the set of nonnegative integers. Lotka-Volterra systems...
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