Center for Machine Perception
Department of Cybernetics, Faculty of Electrical Engineering
Czech Technical University
in
Prague
Methods for Solving Systems of Polynomial
Equations
Zuzana Kúkelová
Zuzana K
ú
kelov
á
kukelova
@cmp.felk.cvut.cz
2
/13
Motivation
Many problems in computer vision can be formulated using systems
of polynomial equations
5 point relative pose problem, 6 point focal length problem
Problem of correcting radial distortion from point correspondences
systems are not trivial
=
> special algorithms have to be designed to
achieve numerical robustness and computational efficiency
Zuzana K
ú
kelov
á
kukelova
@cmp.felk.cvut.cz
3
/13
Polynomial equation in one unknown

Companion matrix
Finding the roots of the polynomial in one unknown, is equivalent to
determining the eigenvalues of so

called companion matrix
The
companion matrix
of the monic polynomial
in one unknown
x
is a
matrix defined as
The characteristic polynomial of
C(p)
is
equal to
p
T
he eigenvalues of
some matrix A
are precisely the roots of
its
characteristic polynomial
Zuzana K
ú
kelov
á
kukelova
@cmp.felk.cvut.cz
4
/13
System of linear polynomial equations

Gauss elimination
System of
n
linear equations in
n
unknowns can be written as
We can represent this system in a matrix form
Zuzana K
ú
kelov
á
kukelova
@cmp.felk.cvut.cz
5
/13
We can rewrite this
Perform Gauss

elimination on the matrix A => reduces the matrix A
to a triangular form
Back

substitution to find the solution of the linear system
System of linear polynomial equations

Gauss elimination
Zuzana K
ú
kelov
á
kukelova
@cmp.felk.cvut.cz
6
/13
System of m polynomial equations in n
unknowns (non

linear)
A system of equations which are given by a set of
m
polynomials in
n
variables with coefficients from
Our goal is to solve this system
Many different methods
Groebner basis methods
Resultant methods
Zuzana K
ú
kelov
á
kukelova
@cmp.felk.cvut.cz
7
/13
System of m polynomial equations in n
unknowns
–
An ideal
An
ideal
generated by polynomials
is the set
of polynomials of the form:
Contains all polynomials we can generate from F
All polynomials in the ideal are zero on solutions of F
Contains an infinite number of polynomials

generators of
Zuzana K
ú
kelov
á
kukelova
@cmp.felk.cvut.cz
8
/13
Groebner basis method w.r.t. lexicographic
ordering
An ideal can be generated by many different sets of generators
which all share the same solutions
Groebner basis w.r.t. the lexicographic ordering which
generates the ideal
I
=
special set of generators which is easy to
solve

one of the generators is a polynomial in one variable only
Zuzana K
ú
kelov
á
kukelova
@cmp.felk.cvut.cz
9
/13
Groebner basis method w.r.t. lexicographic
ordering
A system of initial equations

initial generators of
I
This system of equations can be written in a matrix form
M is the coefficient matrix
X
is the vector of all monomials

is a monomial
Zuzana K
ú
kelov
á
kukelova
@cmp.felk.cvut.cz
10
/13
Groebner basis method w.r.t. lexicographic
ordering
Compute a Groebner basis w.r.t. lexicographic
ordering
Buchberger’s algorithm ~ Gauss elimination
Generator

polynomial in one variable only
Finding the roots of this polynomial using the companion matrix
Back

substitution to find solutions of the whole system
Zuzana K
ú
kelov
á
kukelova
@cmp.felk.cvut.cz
11
/13
An Analogy
Solving system of linear
equations
System of equations
in triangular form
One polynomial equation
in one unknown
Gauss elimination
Solutions
Back

substitution
Solving system of
polynomial equations
Groebner basis
w.r.t lexicographic ordering

One polynomial equation
in one unknown
Buchberger’s algorithm
Solutions
Companion matrix +
Back

substitution
Zuzana K
ú
kelov
á
kukelova
@cmp.felk.cvut.cz
12
/13
System of m polynomial equations in n
unknowns
–
An action matrix
For most problems

“Groebner basis method w.r.t. the lexicographic
ordering” is not feasible (double exponential computational
complexity in general)
Therefore for some problems
A Groebner basis
G
under another ordering, e.g. the graded reverse
lexicographic ordering is constructed
The properties of the
quotient ring
can be used
The “action” matrix of the linear operator of the
multiplication by a polynomial is constructed
The solutions to the set of equations

read off directly from the
eigenvalues and eigenvectors of this matrix
Polynomial
Equations
Gr
öbner
Basis
Action
Matrix
Solutions
Buchberger’s algorithm
Polynomial division
Eigenvectors
Zuzana K
ú
kelov
á
kukelova
@cmp.felk.cvut.cz
13
/13
An Analogy
Solving one Polynomial
Equation in one Unknown
Finding the Eigenvalues
of the Companion Matrix
Compute Companion
Matrix
Solutions
Solving System of
Polynomial Equations
Finding the Eigenvalues
of the Action Matrix
Compute Action Matrix in Quotient Ring A
(Polynomials modulo the Groebner basis)
Solutions
Requires a Groebner Basis
for Input Equations
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