246 Algebraic Operads: An Algorithmic Companion
that is, to determine whether the monomial m belongs to the ideal generated
by the given relations m
0
i
= m
00
i
for i = 1, . . . , k. This easily reduces to a
special case of the ideal membership problem for polynomial rings: determine
whether the monomial m belongs to the ideal generated by the polynomials
m
0
i
−m
00
i
for i = 1, . . . , k. (But note that this reduction only goes one way.) In
an appendix to their paper, Mayr and Meyer give a simplified and corrected
proof of the theorem of Hermann quoted above.
Six years later, Bayer and Stillman [13] published a self-contained and
simplified version of Mayr and Meyer’s example of a polynomial ideal exhibit-
ing doubly exponential degrees for the ideal membership ...