Reduction of everywhere convergent power series with respect to Gröbner bases

Abstract

We introduce a notion of Gröbner 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. Our main theorem states the existence of 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 Gröbner basis then the remainder of that division depends only on the equivalence class of the power series modulo the ideal generated by the polynomials. When the power series which shall be divided is a polynomial the division formula leads to a usual Gröbner representation well known from polynomial rings. Finally, the results are applied 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.