|Title||Reduction of Everywhere Convergent Power Series with respect to Groebner Bases|
|Author(s)|| Joachim Apel, Jürgen Stückrad, Piotr Tworzewski, Tadeusz Winiarski|
|Type||Technical Report, Misc|
|Abstract||In this paper we introduce a notion of Groebner reduction of everywhere convergent power series over the real or complex numbers with respect to ideals generated by polynomials and an admissible term ordering. The presented theory is situated somewhere between the known theories for polynomials and formal power series. The paper results from the cooperation of authors working on different fields of mathematics. So, we hope that the subject of the paper will be interesting for scientists belonging to a width spectrum of research areas, too. For the sake of selfcompleteness we start with a short introduction into the well-known theories of admissible term orderings and Groebner bases for polynomial ideals in section 2. For a comprehensive overview we refer to [BW].|
The third section contains the main theorem which says that there is a formula for the division of everywhere convergent power series over the real or complex numbers by a finite set of polynomials. If the set of polynomials is a Grobner basis then the remainder of that division depends only on the equivalence class of the power series modulo the ideal generated by the Grobner basis. In the case that the power series which shall be divided is even a polynomial the division formula gives a G-representation.
Finally, in section 4 we apply the results to prove the closedness of ideals generated by polynomials in the ring of everywhere convergent power series and to give a very simple proof of the affine version of Serre's graph theorem.