Cyclic4: Comparison of the methods
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Software
We used Bertini to compute a witness superset
for the first-order deflated variety for the Cyclic4 system using
five different methods. The following summarizes these computations.
Three direct approaches
- Dimension-by-dimension algorithm: input
- Cascade algorithm: input
- Regenerative cascade algorithm: input
Diagonal intersection
To perform the diagonal intersection, we first need to decompose
the original system as well as the Jacobian null space system.
- Decompose original system: input
- Decompose Jacobian null space system: input
Since one of the components of the Jacobian null space system is generically
nonreduced, we setup a deflated system. The easiest way to do this is
to print the witness point set and use the following witness set constructor:
- Construct deflated witness set using: input and start
We now perform the intersection for all of the required components using
the following systems.
- Intersection: input
- Intersection using deflated system: input
Regeneration extension
To perform the regeneration extension, we first decompose
the original system.
- Decompose original system: input
Then, we perform the extension on these components using the following system.
- Regeneration extension: input
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