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26 CHAPTER 3. SOLVING POLYNOMIAL SYSTEMS OF EQUATIONS
This chapter is largely based on [47], [48], [50], [81], and [98] and own work [13],
[16], and [20].
3.1 Solving polynomial equations in one variable
For a good understanding of solving systems of polynomial equations, knowledge
about solving univariate polynomial equations is required. The basic subject in this
section is the computation of the zeros of a polynomial of degree d in the variable x:
f(x) =a d x d + a d−1 x d−1 + ...+ a 1 x + a 0 (3.1)
with coefficients a i in the field of real numbers R and a d 0. The variable x ranges
over the field of complex numbers C.
The fundamental theorem of algebra [31] states that the polynomial f(x) has d
roots in the field C of complex numbers counting multiplicities. If the degree d of
f(x) is smaller than or equal to 4, there exist explicit formulas for the roots of f(x)
in terms of the coefficients a 0 ,...,a d , involving square roots and standard arithmetic
operations. One of the main results from Galois Theory states that for univariate
polynomials with a degree larger than 4 such explicit formulas for the roots in terms
of the coefficients do not exist [31].
Numerical root finding can be a challenging task even for the univariate case,
especially when the degree d in (3.1) is high, or when the coefficients a i are given in
floating point expressions or approximations of these.
Numerical methods for solving the univariate polynomial equation f(x) =0can
be based on Newton methods, bisection techniques or sum of squares reformulations
but also on reducing the problem to an eigenvalue problem involving the companion
matrix T x of the polynomial f(x). In this case the companion matrix T x attains the
form:
⎛
⎞
−a
0 0 ... 0
0
a d
−a 1
a d
1 0 ... 0
T x =
0 1 ... 0 .
⎜
⎝ .
.
. .. . . 0 0 ... 1
−a d−1
a d
. (3.2)
⎟
⎠
Note that there are different versions of companion matrices with the coefficients
showing up in the last or first column or the last or first row. The matrix T x is just one
possible companion form which applies to this situation, allowing for an interesting
algebraic interpretation as will be discussed shortly.
Proposition 3.1. The zeros of f(x), including multiplicities, are the eigenvalues of
the matrix T x .
Proposition 3.1 is a standard result in linear algebra. For a proof, see e.g., [98].