# Start Systems

## Total Degree

HomotopyContinuation.total_degreeFunction
total_degree(
F::System;
parameters = nothing,
gamma = cis(2π * rand()),
tracker_options = TrackerOptions(),
endgame_options = EndgameOptions(),
)

Solve the system F using a total degree homotopy. This returns a path tracker (EndgameTracker or OverdeterminedTracker) and an iterator to compute the start solutions. If the system F has declared variable_groups then a multi-homogeneous a start system following [Wam93] will be constructed.

source

## Mixed Volume (Polyhedral Homotopy)

HomotopyContinuation.polyhedralFunction
polyhedral(F::Union{System, AbstractSystem};
only_non_zero = false,
endgame_options = EndgameOptions(),
tracker_options = TrackerOptions())

Solve the system F in two steps: first solve a generic system derived from the support of F using a polyhedral homotopy as proposed in [HS95], then perform a coefficient-parameter homotopy towards F. This returns a path tracker (PolyhedralTracker or OverdeterminedTracker) and an iterator to compute the start solutions.

If only_non_zero is true, then only the solutions with non-zero coordinates are computed. In this case the number of paths to track is equal to the mixed volume of the Newton polytopes of F.

If only_non_zero is false, then all isolated solutions of F are computed. In this case the number of paths to track is equal to the mixed volume of the convex hulls of $supp(F_i) ∪ \{0\}$ where $supp(F_i)$ is the support of $F_i$. See also [LW96].

function polyhedral(
support::AbstractVector{<:AbstractMatrix},
coefficientss::AbstractVector{<:AbstractVector{<:Number}};
kwargs...,
)

It is also possible to provide directly the support and coefficients of the system F to be solved.

Example

We consider a system f which has in total 6 isolated solutions, but only 3 where all coordinates are non-zero.

@var x y
f = System([2y + 3 * y^2 - x * y^3, x + 4 * x^2 - 2 * x^3 * y])
tracker, starts = polyhedral(f; only_non_zero = false)
# length(starts) == 8
count(is_success, track.(tracker, starts)) # 6

tracker, starts = polyhedral(f; only_non_zero = true)
# length(starts) == 3
count(is_success, track.(tracker, starts)) # 3
source

## Overdetermined

HomotopyContinuation.excess_solution_check!Function
excess_solution_check!(path_result::PathResult,
F::RandomizedSystem,
newton_cache = NewtonCache(F.system))

Assigns to the PathResult path_result the return_code :excess_solution if the path_result is a solution of the randomized system F but not of the polynomial system underlying F. This is performed by using Newton's method for non-singular solutions and comparing the residuals of the solutions for singular solutions.

source
• Wam93An efficient start system for multi-homogeneous polynomial continuation, Wampler, C.W. Numer. Math. (1993) 66: 517. https://doi.org/10.1007/BF01385710
• HS95Birkett Huber and Bernd Sturmfels. “A Polyhedral Method for Solving Sparse Polynomial Systems.” Mathematics of Computation, vol. 64, no. 212, 1995, pp. 1541–1555
• LW96T.Y. Li and Xiaoshen Wang. "The BKK root count in C^n". Math. Comput. 65, 216 (October 1996), 1477–1484.