 
Summary: REDUCTION MOD P OF STANDARD BASES
Matthias Aschenbrenner
Department of Mathematics, Statistics,
and Computer Science
University of Illinois at Chicago
851 S. Morgan St. (M/C 249)
Chicago, IL 60607, U.S.A.
maschenb@math.uic.edu.
To Volker Weispfenning, on his 60th birthday.
Abstract
We investigate the behavior of standard bases (in the sense of Hironaka and Grauert)
for ideals in rings of formal power series over commutative rings with respect to special
izations of the coefficients. For instance, we show that any ideal I of the ring of formal
power series A[[X]] = A[[X1, . . . , XN ]] with coefficients in a Noetherian ring A admits
a standard basis whose image under every specialization of A onto a field is a standard
basis of the image of I. Applications include a modular criterion for ideal membership
in Z[[X]] and a constructibility result for ideal membership in K[[X]], where K is a
field.
Partially supported by National Science Foundation grant DMS 0303618.
Keywords: Standard bases, power series, ideal membership, constructibility
