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# 6. Timings and Choice of Examples

We compare the implementation of the sequential algorithm with the standard algerithm , Schreyer's algorithm , LaScala's algorithm and the plain Groebner basis computation .

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
As it is expected the sequential algorithm works especially well on the cyclic examples. Besides this, the algorithm seems to benefit from a great amount of computations in the higher modules of syzygies, whereas it is disadvantageously to have the Groebner basis as the major part of the computation of a free resolution. Our experience shows that the efficiency of the sequential algorithm depends mainly on the fast computation of the extending resolutions . Therefore, it can be expected that a hybrid of the sequential algorithm and the Hilbert-driven algorithm for the extending resolution is the best implementation of the sequential algorithm.

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