6. Timings and Choice of Examples

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

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 ${\bf F}_{32003}$.

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 $I_{n-1}$-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 $L_i$. Therefore, it can be expected that a hybrid of the sequential algorithm and the Hilbert-driven algorithm for the extending resolution $L_i$ is the best implementation of the sequential algorithm.