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Rational Constants of Generic LV Derivations and of Monomial Derivations

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.

Rational functions without poles in a compact set

W. Kucharz (2006)

Colloquium Mathematicae

Let X be an irreducible nonsingular complex algebraic set and let K be a compact subset of X. We study algebraic properties of the ring of rational functions on X without poles in K. We give simple necessary conditions for this ring to be a regular ring or a unique factorization domain.

Rationality of the quotient of ℙ2 by finite group of automorphisms over arbitrary field of characteristic zero

Andrey Trepalin (2014)

Open Mathematics

Let 𝕜 be a field of characteristic zero and G be a finite group of automorphisms of projective plane over 𝕜 . Castelnuovo’s criterion implies that the quotient of projective plane by G is rational if the field 𝕜 is algebraically closed. In this paper we prove that ℙ 𝕜 2 ℙ 𝕜 2 G G is rational for an arbitrary field 𝕜 of characteristic zero.

Real holomorphy rings and the complete real spectrum

D. Gondard, M. Marshall (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

The complete real spectrum of a commutative ring A with 1 is introduced. Points of the complete real spectrum Sper c A are triples α = ( 𝔭 , v , P ) , where 𝔭 is a real prime of A , v is a real valuation of the field k ( 𝔭 ) : = qf ( A / 𝔭 ) and P is an ordering of the residue field of v . Sper c A is shown to have the structure of a spectral space in the sense of Hochster [5]. The specialization relation on Sper c A is considered. Special attention is paid to the case where the ring A in question is a real holomorphy ring.

Recognizing dualizing complexes

Peter Jørgensen (2003)

Fundamenta Mathematicae

Let A be a noetherian local commutative ring and let M be a suitable complex of A-modules. It is proved that M is a dualizing complex for A if and only if the trivial extension A ⋉ M is a Gorenstein differential graded algebra. As a corollary, A has a dualizing complex if and only if it is a quotient of a Gorenstein local differential graded algebra.

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