The timings of Singular are always greater than 0.5 sec, whence, the times
smaller than that are set to be zero. The examples were computed on a
Pentium III 500 MHz with 1GB RAM running under linux, kernel version 2.2.10
As some of the examles are very extensive (random cubics and quintics) we do
not give a complete list of the examples. They can be downloaded (as well as
Singular itself) via anonymous ftp from

www.mathematik.uni-kl.de:/pub/Math/Singular

There is a file test_syz.sing containing a Singular version of the tested
rings and ideals. All examples are global, homogeneous and computed over
the base field
.

At first we give an overview over the sequence of extension of our examples:

Example | Number of generators | Regularity sequence |

Alex4 | 3 | rrn |

standard | 3 | rrr |

f(11,10,3,1) | 3 | rrr |

h(6) | 3 | rrr |

h(7) | 3 | rrr |

g(6,8,10,5,5;0) | 3 | rrr |

f(19,19,4,1) | 3 | rrr |

(max5)2 | 15 | rnnnnnnnnnnnnnn |

randomSyz1 | 4 | rrrr |

randomSyz2 | 5 | rrrrr |

cyclic roots 5 | 5 | rrrrn |

cyclic roots 5 homog | 5 | rrrrr |

Schwarz6 | 6 | rrrrrr |

Kahn4 | 5 | rrrrr |

Iarrobino1 | 15 | rrrnnnnnnnnnnnn |

Random5,3 | 3 | rrr |

Alex1 Mora | 3 | rrn |

Alex 6 | 3 | rrn |

katsura5 | 6 | rrrrrr |

mac1 | 3 | rrn |

mac2 | 3 | rnn |

mac3 | 3 | rrn |

cyclic roots 6 homog | 6 | rrrrnr |

cyclic6 reordered | 6 | rrrrnr |

Here ``r'' stands for a regular while ``n'' stands for a non-regular extension. It can be seen that especially the cyclic examples have an untypical defect compared with other: The last extension is always -regular whereas there are non-regular extensions before. This behaviour results from the homogenization of a constant monomial with a new variable in the last generator. Therefore, it is questionable to use them for more than a computational challenge.

The table of timings looks like follows:

Example | Groebner | Res | Schreyer | LaScala | Seq |

Alex4 | - | 2.1 | - | - | - |

standard | - | 0.7 | - | - | - |

f(11,10,3,1) | - | 0.8 | - | - | - |

h(6) | - | 0.6 | - | - | - |

h(7) | - | 1.0 | - | - | - |

g(6,8,10,5,5;0) | - | - | 0.6 | - | - |

f(19,19,4,1) | - | 17.7 | 1.6 | - | - |

(max5)2 | - | 0.5 | 12.5 | - | - |

randomSyz1 | - | 1.1 | 1.0 | 0.6 | 0.8 |

randomSyz2 | - | 15.9 | 25.7 | 8.2 | 4.5 |

cyclic roots 5 | - | 5.2 | - | - | - |

cyclic roots 5 homog | - | 2000.9 | 0.6 | - | - |

Schwarz6 | - | 0.7 | 1.6 | - | - |

Kahn4 | 0.8 | 27.8 | 67.1 | 52.2 | 28.6 |

Iarrobino1 | - | 4.6 | - | 1.0 | 6.5 |

Random5,3 | 1.4 | 3.4 | 3.4 | 0.7 | 2.7 |

Alex1 Mora | 0.9 | 18.0 | 16.7 | - | 2.2 |

Alex 6 | 1.1 | 25.8 | 9.2 | 0.5 | 2.4 |

katsura5 | - | 9.5 | 3.5 | 5.7 | 0.7 |

mac1 | - | 2.6 | 1.8 | - | - |

mac2 | 7.5 | 58.4 | 86.8 | - | 6.9 |

mac3 | 9.2 | 86.1 | 39.8 | - | - |

cyclic roots 6 homog | - | 29162.5 | 116.4 | 501.4 | 2.1 |

cyclic6 reordered | - | 2482.3 | 1344.6 | 103.0 | 10.7 |