If we just type in /tmp/phc without any option, we run the program in full mode and will pass through all the main menus. A nice application is the verification of the counterexample of Bertrand Haas. We type in haas when the program asks us for the name of the input file. As the output may be rather large, we better save the output file on /tmp. As we run through all the menus, for this system, a good choice is given by the default, so we can type in 0 to answer every question. At the very end, for the output format, it may be good to type in 1 instead of 0, so we can see the progress of the program as it adds solution after solution to the output file.
If we look at the output file for the system in multilin, then we see that the mixed volume equals the 4-homogeneous Bézout number. Since polyhedral methods (e.g. to compute the mixed volume) are computationally more expensive than the solvers based on product homotopies, we can solve the same problem faster. If we run the program on the system in multilin in full mode, we can construct a multi-homogeneous homotopy as follows. At the menu for Root Counts and Method to Construct Start Systems, we type in 1 to select a multi-homogeneous Bézout number. Since there are only 52 possible partitions of a set of four unknowns, it does not take that long for the program to try all 52 partitions and to retain that partition that yields the lowest Bézout number. Once we have this partition, we leave the root counting menu with 0, and construct a linear-product system typing 2 in the menu to construct m-homogeneous start systems. We can save the start system in the file multilin_start (only used for backup). Now we continue just as before.