On the complexity of computing syzygies

Bayer, David; Stillman, Michael

doi:10.1016/s0747-7171(88)80039-7

Cited by 79 publications

(37 citation statements)

References 8 publications

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“…, x r ] for which the ideal membership problem had doubly exponential complexity in terms of the degrees of the generators. Bayer and Stillman [1] showed that these ideals also had doubly exponential regularity, which was exhibited in the first syzygies of a r . In other words t 0 (a r ) = 4 and t 1 (a r ) ≥ 2 2 (r−2)/10 .…”

Section: In Particular If S/i Is Cohen-macaulay Of Codimension C Andmentioning

confidence: 90%

A polynomial bound on the regularity of an ideal in terms of half of the syzygies

McCullough

2012

Mathematical Research Letters

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Abstract. Let K be a field and let S = K[x 1 , . . . , x n ] be a polynomial ring. Consider a homogenous ideal I ⊂ S. Let t i denote reg(Tor S i (S/I, K)), the maximal degree of an ith syzygy of S/I. We prove bounds on the numbers t i for i > n 2 purely in terms of the previous t i . As a result, we give bounds on the regularity of S/I in terms of as few as half of the numbers t i . We also prove related bounds for arbitrary modules. These bounds are often much smaller than the known doubly exponential bound on regularity purely in terms of t 1 .

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Section: In Particular If S/i Is Cohen-macaulay Of Codimension C Andmentioning

confidence: 90%

A polynomial bound on the regularity of an ideal in terms of half of the syzygies

McCullough

2012

Mathematical Research Letters

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“…In this survey we gather recent results which partially answer (2) and (3). We remark that question (1) is still wide open.…”

Section: Introductionmentioning

confidence: 93%

“…In particular, Bayer and Stillman [3,Corollaries 19.11 and 20.21] proved that pd(R/ gin(I)) = pd(R/I) and reg(R/ gin(I)) = reg(R/I). Moreover, the projective dimension of R/ gin(I) can be read off directly from a minimal set of generators as the largest index among the indices of variables appearing in the minimal generators.…”

Section: Background and An Equivalent Problemmentioning

confidence: 99%

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Bounding Projective Dimension

McCullough

Seceleanu

2012

Commutative Algebra

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“…In this field, B Bucherberger, D A Bayer, H Hong, H M Möller, J Gutieerz, L Robbiano, N Takayama, P Loustaunau, T Beker, T Mora, V Weisphenning, W Adams and so on have made many profitable contributions, they have obtained many good results [1][2][3][4][5][6][7]11,12]. Therefore it will be useful to study various properties about Gröbner basis.…”

Section: §1 Introductionmentioning

confidence: 99%