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32 CHAPTER 3. SOLVING POLYNOMIAL SYSTEMS OF EQUATIONS
3.2.3 Subdivision methods
A subdivision method is used to find the roots of a system of polynomial equations
which lie in a specific domain or to isolate real roots. They use an exclusion criterion
to remove a domain if no roots are found to lie inside this domain [85].
By rewriting the set of polynomials as polynomials in the Bernstein form, which
yields a set of linear combinations of Bernstein polynomials B i,j (x) = ( j
i
)
x i (1 − x) j−i
[37], the problem of searching for roots can be reduced to univariate root finding. The
main ingredients of a subdivision solver are a preconditioner to transform the system
of equations into a system which is more numerically stable, a reduction strategy to
reduce the initial domain into a (smaller) search domain and a subdivision strategy,
which is applied iteratively, to subdivide the search domain. The output is a set
of small enough boxes which contain the roots of the original system of polynomial
equations [91].
3.2.4 Algebraic methods
Algebraic methods to solve a system of equations exploit the algebraic relationships
between the unknowns in the system of equations. These methods are mostly based
on Gröbner Basis computations or normal form computations in a quotient algebra.
Gröbner basis computations can be used to solve systems of polynomial equations
by eliminating variables from the system. In this situation the system in Gröbner
basis form has the following upper triangular structure:
⎧
⎪⎨
⎪⎩
g 1(x 1, x 2, x 3, ... , x n)
.
g p(x 1, x 2, x 3, ... , x n)
g p+1(x 2, x 3, ... , x n)
.
g q(x 2, x 3, ... , x n)
g q+1(x 3, ... , x n)
.
.
g r(x n)
.
g s(x n)
(3.14)
This structure can be achieved by using a proper monomial ordering like the lexicographic
ordering. A system with such a triangular structure can be solved in a step
wise order starting with the last (univariate) polynomial: first solve for the variable
x n , then for the variables x n ,x n−1 , then for x n ,x n−1 ,x n−2 and so on, until all the
solutions x 1 ,...,x n of the original system are found. This procedure is also called a
‘back-solve strategy’. The transformation into such a Gröbner basis form is convenient
when using such a procedure for solving a system of equations: one can always work