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

Diagonal intersection
To perform the diagonal intersection, we first need to decompose the original system as well as the Jacobian null space system. 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: We now perform the intersection for all of the required components using the following systems.
Regeneration extension
To perform the regeneration extension, we first decompose the original system. Then, we perform the extension on these components using the following system.

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