Primary decomposition of the Mayr-Meyer ideals

Grete Hermann proved in H] that for any ideal I in an n-dimensional polynomial ring over the eld of rational numbers, if I is generated by polynomials f 1 ; : : : ; f k of degree at most d, then it is possible to write f = P r i f i such that each r i has degree at most deg f + (kd) (2 n). Mayr and Meyer in MM] found (generators) of ideals for which a doubly exponential bound in n is indeed achieved. Bayer and Stillman BS] showed that for these Mayr-Meyer ideals any minimal generating set of syzygies has elements of doubly exponential degree in n. Koh K] modiied the original ideal to obtain homogeneous quadric ideal with doubly exponential syzygies and ideal membership equations. This paper examines the primary decomposition structure of these ideals. The motivation for this paper came from some questions raised by Bayer, Huneke and Stillman about these ideals: is the doubly exponential behavior due to the number of minimal and/or associated primes, or to the nature of one of them? There exist algorithms for computing primary decompositions of ideals (see Gianni-Trager-Zacharias GTZ], Eisenbud-Huneke-Vasconcelos EHV], or Shimoyama-Yokoyama SY]), and they have been implemented on the symbolic computer algebra program Singular and partially on Macaulay2. However, the Mayr-Meyer ideals have variable degree and a variable number of variables over an arbitrary eld, and there are no algorithms to deal with this generality. Thus any primary decomposition of the Mayr-Meyer ideals has to be accomplished with traditional proof methods. Small cases were partially veriied on Macaulay2 and Singular, and the emphasis here is on \partially": the computers available to me quickly run out of memory. The Mayr-Meyer ideals are binomial, so by Eisenbud-Sturmfels ES] all the associated primes themselves are also binomial ideals. It turns out that many associated primes are even monomial, which simpliies many of the calculations. All the minimal components are found and proved in Section 2. However, my attempts at nding the embedded components became notationally and computationally unwieldy (see http://math.nmsu.edu/~iswanson for these and other computations with the Mayr-Meyer ideals not included here), so instead I tried to nd only the embedded primes, not necessarily their components. The main tool used below for this are various short exact sequences, and the fact that the associated primes of the middle module in a short exact sequence is contained …