RootCounter

Originally, this code was and currently still is available in the ToricVarieties_project. To make this tool available to other computer algebra systems, the code has been moved to this repository at the beginning of 2023. For more background, see e.g. this article and references therein.

Code coverage

General tests

Description	Test result	Runtime (min)
Basic tests		2
Simple root counting		2

Root counting for QSMs

Polytope	$\overline{K}_{B_3}^3$	Elliptic curve	Geometric multiplicity $\mu$	Total number of roots	Estimate for number of triangulations $T$	Upper bound for number of configurations	Runtime (min)	Test result
8	6	✔️	$12^3$	$12^8$	$3.867 \times 10^{13} \leq T \leq 2.828 \times 10^{16}$	$10^{24}$	2
4	6	❌	$12^4$	$12^8$	$3.188 \times 10^{11}$	$10^{20}$	2
134	6	❌	$12^4$	$12^8$	$7.538 \times 10^{10}$	$10^{19}$	2
128, 130, 136, 236	6	❌	$12^4$	$12^8$	$3.217 \times 10^{11}$	$10^{20}$	1
88	10	✔️	$20^5$	$20^{12}$	$5.231 \times 10^{10} \leq T \leq 1.246 \times 10^{14}$	$10^{30}$	3
110	10	✔️	$20^5$	$20^{12}$	$5.239 \times 10^{8}$	$10^{24}$	4
272, 274	10	✔️	$20^5$	$20^{12}$	$3.212 \times 10^{12} \leq T \leq 2.481 \times 10^{15}$	$10^{31}$	4
387	10	✔️	$20^5$	$20^{12}$	$6.322 \times 10^{10} \leq T \leq 6.790 \times 10^{12}$	$10^{28}$	9
798, 808, 810, 812	10	✔️	$20^5$	$20^{12}$	$1.672 \times 10^{10} \leq T \leq 2.515 \times 10^{13}$	$10^{29}$	12
254	10	❌	$20^6$	$20^{12}$	$1.568 \times 10^{10}$	$10^{26}$	6
52	10	❌	$20^6$	$20^{12}$	$1.248 \times 10^{8}$	$10^{24}$	7
302	10	❌	$20^6$	$20^{12}$	$5.750 \times 10^{7}$	$10^{23}$	3
786	10	❌	$20^6$	$20^{12}$	$9.810 \times 10^{8}$	$10^{24}$	74
762	10	❌	$20^6$	$20^{12}$	$1.087 \times 10^{11} \leq T \leq 2.854 \times 10^{13}$	$10^{30}$	35
417	10	❌	$20^6$	$20^{12}$	$1.603 \times 10^{9}$	$10^{25}$	34
838	10	❌	$20^6$	$20^{12}$	$4.461 \times 10^{9}$	$10^{25}$	67
782	10	❌	$20^6$	$20^{12}$	$3.684 \times 10^{9}$	$10^{25}$	124
377, 499, 503	10	❌	$20^6$	$20^{12}$	$4.461 \times 10^{9}$	$10^{25}$	98
1348	10	❌	$20^6$	$20^{12}$	$4.285 \times 10^{9}$	$10^{25}$	239
882, 856	10	❌	$20^6$	$20^{12}$	$3.180 \times 10^{9}$	$10^{25}$	96
1340	10	❌	$20^6$	$20^{12}$	$4.496 \times 10^{9}$	$10^{25}$	> 6 hours	❌
1879	10	❌	$20^6$	$20^{12}$	$4.461 \times 10^{9}$	$10^{25}$	> 6 hours	❌
1384	10	❌	$20^6$	$20^{12}$	$7.040 \times 10^{9}$	$10^{25}$	> 6 hours	❌

(Runtimes are taken from #28.)