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# Appendix A: Benchmark examples

I this appendix, we show some more details about the sets of polynomial we used for our GB computation test runs.

Table 3: Summary of properties of benchmark examples
 Example #vars #polys homog Degs Deg Ref ecyclic 7 43 7 no 7 27 ecyclic 6 31 6 no 6 17 rcyclic i no homog 2mat3 19 8 yes 4 13 [14] 2mat3 18 8 no 4 13 homog gonnet 18 19 yes 2 11 [6] gonnet 17 19 no 2 11 [6] schwarz 11 11 11 no 2 13 schwarz 10 10 10 no 2 12 katsura 8 9 9 no 2 10 [13] katsura 7 8 8 no 2 9 [13] bjork 8 8 9 no 8 18 [5] homog cyclic 7 8 7 yes 7 20 [5] cyclic 7 7 7 no 7 27 [5] homog cyclic 6 7 6 yes 6 17 [5] cyclic 6 6 6 no 6 17 [5] homog alex 3 6 4 yes 14 51 alex 3 5 4 no 14 51 gerhard 1 5 3 yes 10 32 symmetric 6 5 5 yes 6 23 [12] homog alex 2 5 3 yes 12 40 cohn2 4 4 no 6 20 [14] alex 2 4 3 no 12 33 gerhard 2 4 3 yes 9 44 gerhard 3 4 3 yes 23 81

Table 3 lists a summary of their properties: column #vars shows the number of occurring variables, column #polys the number of elements (polynomials), column homog gives the homogeneity, and Degs shows the maximal degree of the input sets. Deg gives the maximal degree occurring during the GB computation w.r.t. the degree reverse lexicographical ordering. The last column gives references to, our sources of these examples. Those without a reference are from the collection of examples of the SINGULAR team.

Finally, in the rest of this appendix, we completely list the all the used examples.

cyclic n:
, n generators pk:

For example, cyclic 4:

homog cyclic n:
, n generators pk:

For example, homog cyclic 4:

rcyclic n:
, n generators pk:

For example, rcyclic 4:

ecyclic n:
, n generators pk:

For example, ecyclic 4:

[ht]

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