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$_{\max}$	Ref
ecyclic 7	43	7	no	7	27
ecyclic 6	31	6	no	6	17
rcyclic $10 \!\leq\! i\!\leq\! 19$	i	$i\!-\!1$	no	$i\!-\!1$	$\lfloor i/2 \rfloor\!*\!2\!+\!4$
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 Deg_s shows the maximal degree of the input sets. Deg$_{\max}$ 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:

$[x_1, \ldots x_n]$, n generators p_k:

$$1 \leq k < n: p_k = \sum_{j=1}^n \prod_{i=j}^{j+k-1} x_i, \quad p_n = \prod_{i=1}^{n} x_i -1$$

For example, cyclic 4:

$$\left( \begin{array}{c} x_{1} + x_{2} + x_{3} + x_{4} \\ x_{... ..._{4}x_{1}x_{2} \\ x_{1}x_{2}x_{3}x_{4} - 1 \end{array}\right)$$

homog cyclic n:

$[x_1, \ldots x_{n+1}]$, n generators p_k:

$$1 \leq k < n: p_k = \sum_{j=1}^n \prod_{i=j}^{j+k-1} x_i, \quad p_n = \prod_{i=1}^{n} x_i - x_{n+1}^n$$

For example, homog cyclic 4:

$$\left( \begin{array}{c} x_{1} + x_{2} + x_{3} + x_{4} \\ x_{... ...x_{1}x_{2} \\ x_{1}x_{2}x_{3}x_{4} - x_5^4 \end{array}\right)$$

rcyclic n:

$[x_1, \ldots x_n]$, n generators p_k:

$$1 \leq k < n: p_k = \sum_{j=1}^3 \prod_{i=j}^{j+k-1} x_i, \quad p_n = \prod_{i=1}^{n} x_i -1$$

For example, rcyclic 4:

$$\left( \begin{array}{c} x_{1} + x_{2} + x_{3} \\ x_{1}x_{2} ... ..._{3}x_{4}x_{1} \\ x_{1}x_{2}x_{3}x_{4} - 1 \end{array}\right)$$

ecyclic n:

$[x_1, \ldots x_{n(n-1) + 1}]$, n generators p_k:

$$1 \!\leq\! k \!<\! n: p_k \!=\! \sum_{j=1}^n \prod_{i=j}^{j+k... ...x_{(j-1)n+i}, \quad p_n \!= \! \prod_{i=1}^{n} x_{(i-1)n + 1}$$

For example, ecyclic 4:

$$\left( \begin{array}{c} x_{1} + x_{2} + x_{3} + x_{4} \\ x_{... ..._{4}x_{8}x_{12}\\ x_{1}x_{5}x_{9}x_{13} - 1 \end{array}\right)$$

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