The minimal components of the Mayr–Meyer ideals

Abstract

rifi such that each ri has degree at most deg f+(kd) n). Mayr and Meyer in [MM] found (generators) of a family 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] modified the original ideals to obtain homogeneous quadric ideals with doubly exponential syzygies and ideal membership equations. Bayer, Huneke and Stillman asked whether the doubly exponential behavior is due to the number of minimal and/or associated primes, or to the nature of one of them? This paper examines the minimal components and minimal primes of the Mayr-Meyer ideals. In particular, in Section 2 it is proved that the intersection of the minimal components of the Mayr-Meyer ideals does not satisfy the doubly exponential property, so that the doubly exponential behavior of the Mayr-Meyer ideals must be due to the embedded primes. The structure of the embedded primes of the Mayr-Meyer ideals is examined in [S2]. There exist algorithms for computing primary decompositions of ideals (see GianniTrager-Zacharias [GTZ], Eisenbud-Huneke-Vasconcelos [EHV], or Shimoyama-Yokoyama [SY]), and they have been partially implemented on the symbolic computer algebra programs Singular and Macaulay2. However, the Mayr-Meyer ideals have variable degree and a variable number of variables over an arbitrary field, 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 of the primary decomposition analysis were partially verified on Macaulay2 and Singular, and the emphasis here is on “partially”: the computers quickly run out of memory.