Primitivity in Differential Operator Rings.
K.R. Goodearl, R.B. Jr. Warfield (1982)
Mathematische Zeitschrift
Andrzej Nowicki (1982)
Colloquium Mathematicae
Andrzej Nowicki (1978)
Colloquium Mathematicae
Janusz Zieliński (2013)
Bulletin of the Polish Academy of Sciences. Mathematics
We describe the fields of rational constants of generic four-variable Lotka-Volterra derivations. Thus, we determine all rational first integrals of the corresponding systems of differential equations. Such systems play a role in population biology, laser physics and plasma physics. They are also an important part of derivation theory, since they are factorizable derivations. Moreover, we determine the fields of rational constants of a class of monomial derivations.
Gene Freudenburg (2001)
Annales Polonici Mathematici
We give an overview of recent results concerning kernels of triangular derivations of polynomial rings. In particular, we examine the question of finite generation in dimensions 4, 5, 6, and 7.
Marius Van der Put (1997/1998)
Séminaire Bourbaki
O. Villamayor U. (2008)
Revista Matemática Iberoamericana
Ernst Kunz, Reinhold Hübl (1990)
Journal für die reine und angewandte Mathematik
Masumi Kersken (1984)
Manuscripta mathematica
Reinhold Hübl (1994)
Mathematische Annalen
Dayan Liu, Xiaosong Sun (2022)
Czechoslovak Mathematical Journal
Let be a field of characteristic zero and a -domain. Let be a retract of being the kernel of a locally nilpotent derivation of . We show that if for some principal ideal (in particular, if is a UFD), then , i.e., is a polynomial algebra over in one variable. It is natural to ask that, if a retract of a -UFD is the kernel of two commuting locally nilpotent derivations of , then does it follow that ? We give a negative answer to this question. The interest in retracts comes...
Janusz Zieliński (2013)
Open Mathematics
Lotka-Volterra systems appear in population biology, plasma physics, laser physics and derivation theory, among many others. We determine the rings of constants of four-variable Lotka-Volterra derivations with four parameters C 1, C 2, C 3, C 4 ∈ k, where k is a field of characteristic zero. Thus, we give a full description of polynomial first integrals of the respective systems of differential equations.
Janusz Zieliński, Piotr Ossowski (2013)
Czechoslovak Mathematical Journal
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...
Gail Letzter, Leonid Makar-Limanov (1990)
Bulletin de la Société Mathématique de France
G. J. Heckman, E. M. Opdam (1987)
Compositio Mathematica
G. J. Heckman (1987)
Compositio Mathematica
Masumi Kersken, Uwe Storch (1990)
Banach Center Publications
Norihiro Wada (2011)
Colloquium Mathematicae
Let k be a field and k[x,y] the polynomial ring in two variables over k. Let D be a higher k-derivation on k[x,y] and D̅ the extension of D on k(x,y). We prove that if the kernel of D is not equal to k, then the kernel of D̅ is equal to the quotient field of the kernel of D.
William F. Keigher (1983)
Cahiers de Topologie et Géométrie Différentielle Catégoriques
Andrzej Nowicki (1984)
Colloquium Mathematicae