Primary decomposition of the Mayr-Meyer ideals, preprint
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h0i = c0i s ? fbdi ; i = 1; 2; 3; 4; 0
then ten level 1 generators:
H11 = s1 ? sc01; H12 = f1 ? sc04; h13 = fc01 ? sc02; h14 = fc04 ? sc03; h15 = s (c03 ? c02 ) ; h16 = f (c02b01 ? c03 b04) ; h1;6+i = fc02c1i (b02 ? b1i b03) ; i = 1; : : :; 4;
1
module. Removing the redundant prime ideals is a much harder process, and takes up most of Sections 4 and 5. In Section 6, the search for the associated primes of the Mayr-Meyer ideals is reduced to the search for the associated primes of a recursively de ned sequence of ideals. The associated primes of these recursively de ned ideals are analyzed in Section 7. All of the primes obtained in Sections 1 through 6 are indeed associated to the Mayr-Meyer ideals, but it may be that some of the primes obtained in Section 7 are not associated to the Mayr-Meyer ideals. The elimination of redundant primes gets progressively harder for primes containing more and more elements, and is not completed here. The Mayr-Meyer ideals depend on two parameters, n and d, where the number of variables in the ring is O(n) and the degree of the given generators of the ideal is O(d). Both n and d are positive integers. The primary decomposition in the case n = 1 is very di erent from the case n 2, and is given in S]. In this paper it is always assumed that n 2. Here is the de nition of the Mayr-Meyer ideals: let n; d 2 be integers and K a eld. Let s; f; sr+1; fr+1; br1; br2; br3; br4; cr1 ; cr2; cr3; cr4 be variables over K , with r = 0; 1; : : :; n ? 1. (This notation closely follows that of K].) Set
the rst six level r generators, r = 2; : : :; n:
Hr1 = sr ? sc01 cr?1;1; Hr2 = fr ? sc01 cr?2;1cr?1;4 ; hr3 = sc01c11 cr?3;1 (cr?2;4 cr?1;1 ? cr?2;1 cr?1;2) ; hr4 = sc01c11 cr?3;1 (cr?2;4 cr?1;4 ? cr?2;1 cr?1;3) ; hr5 = sc01c11 cr?2;1 (cr?1;3 ? cr?1;2 ) ; hr6 = sc01c11 cr?3;1 cr?2;4 (cr?1;2br?1;1 ? cr?1;3 br?1;4) ;
Primary decomposition of the Mayr-Meyer ideals
Irena Swanson November 6, 2001
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 f1; : : : ; fk of degree P at most d, then it is possible to write f = ri fi such that each ri has degree at most deg f + (kd)(2 ) . 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] modi ed 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 GianniTrager-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 veri ed 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 simpli es 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 /~iswanson for these and other computations with the MayrMeyer 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 in the union of the associated primes of the two other modules (see Sections 4-7). However, not all the primes in this union need to be associated to the middle
2thBiblioteka last four level r generators, r = 2; :::; n ? 1:
n
The author thanks the NSF for partial support on grants DMS-0073140 and DMS9970566. 1991 Mathematics Subject Classi cation. 13C13, 13P05
Key words and phrases. Primary decomposition, Mayr-Meyer.
S = K s; f; sr+1; fr+1; bri ; crijr = 0; : : :; n ? 1; i = 1; : : :; 4]:
Thus S is a polynomial ring of dimension 10n + 2. The following generators de ne the Mayr-Meyer ideal M (n; d): rst the four level 0 generators:
h0i = c0i s ? fbdi ; i = 1; 2; 3; 4; 0
then ten level 1 generators:
H11 = s1 ? sc01; H12 = f1 ? sc04; h13 = fc01 ? sc02; h14 = fc04 ? sc03; h15 = s (c03 ? c02 ) ; h16 = f (c02b01 ? c03 b04) ; h1;6+i = fc02c1i (b02 ? b1i b03) ; i = 1; : : :; 4;
1
module. Removing the redundant prime ideals is a much harder process, and takes up most of Sections 4 and 5. In Section 6, the search for the associated primes of the Mayr-Meyer ideals is reduced to the search for the associated primes of a recursively de ned sequence of ideals. The associated primes of these recursively de ned ideals are analyzed in Section 7. All of the primes obtained in Sections 1 through 6 are indeed associated to the Mayr-Meyer ideals, but it may be that some of the primes obtained in Section 7 are not associated to the Mayr-Meyer ideals. The elimination of redundant primes gets progressively harder for primes containing more and more elements, and is not completed here. The Mayr-Meyer ideals depend on two parameters, n and d, where the number of variables in the ring is O(n) and the degree of the given generators of the ideal is O(d). Both n and d are positive integers. The primary decomposition in the case n = 1 is very di erent from the case n 2, and is given in S]. In this paper it is always assumed that n 2. Here is the de nition of the Mayr-Meyer ideals: let n; d 2 be integers and K a eld. Let s; f; sr+1; fr+1; br1; br2; br3; br4; cr1 ; cr2; cr3; cr4 be variables over K , with r = 0; 1; : : :; n ? 1. (This notation closely follows that of K].) Set
the rst six level r generators, r = 2; : : :; n:
Hr1 = sr ? sc01 cr?1;1; Hr2 = fr ? sc01 cr?2;1cr?1;4 ; hr3 = sc01c11 cr?3;1 (cr?2;4 cr?1;1 ? cr?2;1 cr?1;2) ; hr4 = sc01c11 cr?3;1 (cr?2;4 cr?1;4 ? cr?2;1 cr?1;3) ; hr5 = sc01c11 cr?2;1 (cr?1;3 ? cr?1;2 ) ; hr6 = sc01c11 cr?3;1 cr?2;4 (cr?1;2br?1;1 ? cr?1;3 br?1;4) ;
Primary decomposition of the Mayr-Meyer ideals
Irena Swanson November 6, 2001
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 f1; : : : ; fk of degree P at most d, then it is possible to write f = ri fi such that each ri has degree at most deg f + (kd)(2 ) . 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] modi ed 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 GianniTrager-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 veri ed 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 simpli es 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 /~iswanson for these and other computations with the MayrMeyer 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 in the union of the associated primes of the two other modules (see Sections 4-7). However, not all the primes in this union need to be associated to the middle
2thBiblioteka last four level r generators, r = 2; :::; n ? 1:
n
The author thanks the NSF for partial support on grants DMS-0073140 and DMS9970566. 1991 Mathematics Subject Classi cation. 13C13, 13P05
Key words and phrases. Primary decomposition, Mayr-Meyer.
S = K s; f; sr+1; fr+1; bri ; crijr = 0; : : :; n ? 1; i = 1; : : :; 4]:
Thus S is a polynomial ring of dimension 10n + 2. The following generators de ne the Mayr-Meyer ideal M (n; d): rst the four level 0 generators: