The First Mayr - Meyer

This paper gives a complete primary decomposition of the rst, that is, the smallest, Mayr-Meyer ideal, its radical, and the intersection of its minimal components. The particular membership problem which makes the Mayr-Meyer ideals' complexity doubly exponential in the number of variables is here examined also for the radical and the intersection of the minimal components. It is proved that for the rst Mayr-Meyer ideal the complexity of this membership problem is the same as for its radical. This problem was motivated by a question of Bayer, Huneke and Stillman. 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 , where each r i has degree at most deg f + (kd) (2 n). Mayr and Meyer in MM] found ideals J(n; d) for which a doubly exponential bound in n is indeed achieved. Bayer and Stillman BS] showed that for these same ideals also any minimal generating set of syzygies has elements of degree which is doubly exponential in n. Koh K] modiied the original ideal to obtain homogeneous quadric ideals with doubly exponential degrees of syzygies and ideal membership coeecients. Bayer, Huneke and Stillman have raised questions about the structure of these Mayr-Meyer ideals: is the doubly exponential behavior due to the number of minimal primes, to the number of associated primes, or to the structure of one of them? This paper, together with S], is an attempt at answering these questions. More precisely, the Mayr-Meyer ideal J(n; d) is an ideal in a polynomial ring in 10n + 2 variables whose generators have degree at most d + 2. This paper analyzes the case n = 1 and shows that in this base case the embedded components do not play a role. Theorem 1 of this paper gives a complete primary decomposition of J(1; d), after which the intersection of the minimal components and the radical come as easy corollaries. The last proposition shows that the complexity of the particular membership problem from 1 MM, BS, K] for the radical of J(1; d) is the same as the complexity of the membership problem for J(1; d). Thus at least for the case n …