# Start Systems

## Total Degree

`HomotopyContinuation.total_degree`

— Function```
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.

## Mixed Volume (Polyhedral Homotopy)

`HomotopyContinuation.polyhedral`

— Function```
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
```

`HomotopyContinuation.PolyhedralTracker`

— Type`PolyhedralTracker <: AbstractPathTracker`

This tracker realises the two step approach of the polyhedral homotopy. See also [`polyhedral`

].

## Overdetermined

`HomotopyContinuation.OverdeterminedTracker`

— Type`OverdeterminedTracker(tracker::AbstractPathTracker, F::RandomizedSystem)`

Wraps the given `AbstractPathTracker`

`tracker`

to apply `excess_solution_check`

for the given randomized system `F`

on each path result.

`HomotopyContinuation.square_up`

— Function`square_up(F::Union{System, AbstractSystem})`

Creates the `RandomizedSystem`

$\mathfrak{R}(F(x); N)$ where $N$ is the number of variables of `F`

.

`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.

`HomotopyContinuation.excess_solution_check`

— Function`excess_solution_check(F::RandomizedSystem)`

Returns a function `λ(::PathResult)`

which performs the excess solution check. The call `excess_solution_check(F)(path_result)`

is identical to `excess_solution_check!(F, path_result)`

. See also `excess_solution_check!`

.

- 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.