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
