544 research outputs found
Free integro-differential algebras and Groebner-Shirshov bases
The notion of commutative integro-differential algebra was introduced for the algebraic study of boundary problems for linear ordinary differential equations. Its noncommutative analog achieves a similar purpose for linear systems of such equations. In both cases, free objects are crucial for analyzing the underlying algebraic structures, e.g. of the (matrix) functions.
In this paper we apply the method of Groebner-Shirshov bases to construct the free (noncommutative) integro-differential algebra on a set. The construction is from the free Rota-Baxter algebra on the free differential algebra on the set modulo the differential Rota-Baxter ideal generated by the noncommutative integration by parts formula. In order to obtain a canonical basis for this quotient, we first reduce to the case when the set is finite. Then in order to obtain the monomial order needed for the Composition-Diamond Lemma, we consider the free Rota-Baxter algebra on the truncated free differential algebra. A Composition-Diamond Lemma is proved in this context, and a Groebner-Shirshov basis is found for the corresponding differential Rota-Baxter ideal
Complex Centers of Polynomial Differential Equations
We present some results on the existence and nonexistence of centers for
polynomial first order ordinary differential equations with complex
coefficients. In particular, we show that binomial differential equations
without linear terms do not complex centers. Classes of polynomial differential
equations, with more than two terms, are presented that do not have complex
centers. We also study the relation between complex centers and the Pugh
problem. An algorithm is described to solve the Pugh problem for equations
without complex centers. The method of proof involves phase plane analysis of
the polar equations an a local study of periodic solutions.Comment: 18 page
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