Download 3 Basics of Polynomial Theory

3
Basics of Polynomial Theory
3.1 Polynomial Equations
In geodesy and geoinformatics, most observations are related to unknowns parameters through equations of algebraic (polynomial) type.
In cases where the observations are not of polynomial type, as exemplified by the GPS meteorology problem of Chap. 13, they are converted
via Theorem 3.1 on p. 20 into polynomials. The unknown parameters
are then be obtained by solving the resulting polynomial equations.
Such solutions are only possible through application of operations addition and multiplication on polynomials which form elements of polynomial rings. This chapter discusses polynomials and the properties
that characterize them. Starting from the definitions of monomials,
basic polynomial aspects that are relevant for daily operations are presented. A monomial is defined as
Definition 3.1 (Monomial). A monomial is a multivariate product
of the form xα1 1 xα2 2 . . .xαnn , (α1 , . . ., αn ) ∈ Zn+ in the variables x1 , . . ., xn .
In Definition 3.1 above, the set Zn+ comprises positive elements of the
set of integers (2.2) that we saw in Chap. 2, p. 8.
Example 3.1 (Monomial). Consider the system of equations for solving
distances in the three-dimensional resection problem given as (see e.g.,
(11.44) on p. 180)
 2
x1 + 2a12 x1 x2 + x22 + ao = 0
 x22 + 2b23 x2 x3 + x23 + bo = 0

 x23 + 2c31 x3 x1 + x21 + co = 0
where x1 ∈ R+ , x2 ∈ R+ , x3 ∈ R+ .
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3 Basics of Polynomial Theory
The variables {x1 , x2 , x3 } are unknowns ©while the other terms areª
known constants. The products of variables x21 , x1 x2 , x22 , x2 x3 , x23 , x3 x1
are monomials in {x1 , x2 , x3 }.
Summation of monomials form polynomials defined as
Definition 3.2 (Polynomial). A polynomial f ∈ k [x1 , . . . , xn ] in
variables x1 , . . . , xn with coefficients in the field k is a finite linear
combination of monomials with pairwise different terms expressed as
f=
X
α
aα xα , aα ∈ k, xα = (xα1 , . . ., xαn ), α = (α1 , . . ., αn ), (3.1)
where aα are coefficients in the field k, e.g., R or C and xα the monomials.
Example 3.2 (Polynomials). Equations
 2
x1 + 2a12 x1 x2 + x22 + ao = 0
 x22 + 2b23 x2 x3 + x23 + bo = 0
x23 + 2c31 x3 x1 + x21 + co = 0,
in Example 3.1 are multivariate polynomials. The first expression is a
multivariate polynomial
{x1 , x2 } and a linear combiª
© in two variables
nation of monomials x21 , x1 x2 , x22 . The second expression is a multivariate polynomial© in two variables
{x2 , x3 } and a linear combination
ª
of the monomials x22 , x2 x3 , x23 , while the third expression is a multivariate polynomial© in two variables
{x3 , x1 } and a linear combination
ª
2
2
of the monomials x3 , x3 x1 , x1 .
In Example 3.2, the coefficients of the polynomials are elements of
the set Z. In general, the coefficients can take on any sets Q, R, C of
number rings or other rings such as modular arithmetic rings. These
coefficients can be added, subtracted, multiplied or divided, and as such
play a key role in determining the solutions of polynomial equations.
The definition of the set to which the coefficients belong determines
whether a polynomial equation is solvable or not. Consider the following
example:
Example 3.3. Given an equation 9w2 − 1 = 0 with the coefficients in
the integral domain, obtain the integer solutions. Since the coefficient
9 ∈ Z, the equation does not have a solution. If instead the coefficient
1
9 ∈ Q, then the solution w = ± exist.
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3.2 Polynomial Rings
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From Definition 2.1 of algebraic, polynomials become algebraic once
(3.1) is equated to 0. The fundamental problem of algebra can thus be
stated as the solution of equations of form (3.1) equated to 0.
3.2 Polynomial Rings
In Sect. 2.3 of Chap. 2, the theory of rings was introduced with respect
to numbers. Apart from the number rings, polynomials are objects
that also satisfy ring axioms leading to “polynomial rings” upon which
operations “addition” and “multiplication” are implemented.
3.2.1 Polynomial Objects as Rings
Polynomial rings are defined as
Definition 3.3 (Polynomial ring). Consider a ring R say of real
numbers R. Given a variable x ∈
/ R, a univariate polynomial f (x) is
formed (see Definition 3.2 on p. 18) by assigning coefficients ai ∈ R
to the variable and obtaining summation over finite number of distinct
integers. Thus
X
f (x) =
cα xα , cα ∈ R, α ≥ 0
α
is said to be a univariatePpolynomial over R P
. If two polynomials are
dj xj , then two binary
ci xi and f2 (x) =
given such that f1 (x) =
j
i
operations “addition” and “multiplication” can be defined on these polynomials such that:
P
(a) Addition: f1 (x) + f2 (x) = ek xk , ek = ck + dk , ek ∈ R
k P
P
ci dj , gk ∈ R .
(b) Multiplication: f1 (x).f2 (x) = gk xk , gk =
k
i+j=k
A collection of polynomials with these “additive” and “multiplicative”
rules form a commutative ring with zero element and identity 1. A univariate polynomial f (x) obtained by assigning elements ci belonging to
the ring R to the variable x is called a polynomial ring and is expressed
as f (x) = R[x]. In general the entire collection of all polynomials in
x1 , . . . , xn , with coefficients in the field k that satisfy the definition of
a ring above are called a polynomial rings.
Designated P, polynomial rings are represented by n unknown variables
xi over k expressed as P := k [x1 , . . ., xn ] . Its elements are polynomials
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3 Basics of Polynomial Theory
known as univariate when n = 1 and multivariate otherwise. The distinction between a polynomial ring and a polynomial is that the latter
is the sum of a finite set of monomials (see e.g., Definition 3.1 on p. 17)
and is an element of the former.
Example 3.4. Equations
 2
x1 + 2a12 x1 x2 + x22 + ao = 0
 x22 + 2b23 x2 x3 + x23 + bo = 0
x23 + 2c31 x3 x1 + x21 + co = 0
of Example 3.1 are said to be polynomials in three variables [x1 , x2 , x3 ]
forming elements of the polynomial ring P over the field of real numbers
R expressed as P := R [x1 , x2 , x3 ].
Polynomials that we use in solving unknown parameters in various problems, as we shall see later, form elements of polynomial rings.
Polynomial rings provide means and tools upon which to manipulate
the polynomial equations. They can either be added, subtracted, multiplied or divided. These operations on polynomial rings form the basis
of solving systems of equations algebraically as will be made clear in the
chapters ahead. Next, we state the theorem that enables the solution
of nonlinear systems of equations in geodesy and geoinformatics.
Theorem 3.1. Given n algebraic (polynomial) observational equations,
where n is the dimension of the observation space Y of order l in m unknown variables , and m is the dimension of the parameter space X, the
application of least squares solution (LESS) to the algebraic observation
equations gives (2l − 1) as the order of the set of nonlinear algebraic
normal equations. There exists m normal equations of the polynomial
order (2l − 1) to be solved.
Proof. Given nonlinear algebraic equations fi ∈ k{ξ1 , . . . , ξm } expressed as

f1 ∈ k{ξ1 , . . . , ξm }
 f2 ∈ k{ξ1 , . . . , ξm }


.

(3.2)

.


.
fn ∈ k{ξ1 , . . . , ξm },
with the order considered as l, we write the objective function to be
minimized as
3.2 Polynomial Rings
kf k2 = f12 + . . . . + fn2 | ∀fi ∈ k{ξ1 , . . . , ξm },
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(3.3)
and obtain the partial derivatives (first derivatives of 3.3) with respect
to the unknown variables {ξ1 , . . . , ξm }. The order of (3.3) which is
l2 then reduces to (2l − 1) upon differentiating the objective function
with respect to the variables ξ1 , . . . , ξm . Thus resulting in m normal
equations of the polynomial order (2l − 1).
u
t
Example 3.5 (Pseudo-ranging problem). For pseudo-ranging or distance
equations, the order of the polynomials in the algebraic observational
equations is l = 2. If we take the “pseudo-ranges squared” or “distances
squared”, a necessary procedure in-order to make the observation equations “algebraic” or “polynomial”, and implement least squares solution
(LESS), the objective function which is of order l = 4 reduces by one
to order l = 3 upon differentiating once. The normal equations are of
order l = 3 as expected.
The significance of Theorem 3.1 is that all observational equations of
interest are successfully converted to “algebraic” or “polynomial” equations. This implies that problems requiring exact algebraic solutions
must first have their equations converted into algebraic. This will be
made clear in Chap. 13 where trigonometric nonlinear system on equations are first converted into algebraic.
3.2.2 Operations “Addition” and “Multiplication”
Definition 3.3 implies that a polynomial ring qualifies as a ring based
on the applications of operations “addition” and “multiplication” on
its coefficients. In this case, the axioms that follow the Abelian group
with respect to “addition” and the semi group with respect to “multiplication” readily follow. Of importance in manipulating polynomial
rings using operations “addition” and “multiplication” is the concept
of division of polynomials defined as
Definition 3.4 (Polynomial division). Consider the polynomial ring
k[x] whose elements are polynomials f (x) and g(x). There exists unique
polynomials p(x) and r(x) also elements of polynomial ring k[x] such
that
f (x) = g(x)p(x) + r(x),
with either r(x) = 0 or degree of r(x) is less than the degree of g(x).
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3 Basics of Polynomial Theory
For univariate polynomials, as in Definition 3.4, the Euclidean algorithm employs operations “addition” and “multiplication” to factor
polynomials in-order to reduce them to satisfy the definition of division
algorithm.
3.3 Factoring Polynomials
In-order to understand the factorization of polynomials, it is essential
to revisit some of the properties of prime numbers of integers. This is
due to the fact that polynomials behave much like integers. Whereas
for integers, any integer n > 1 is either prime (i.e., can only be factored by 1 and n itself) or a product of prime numbers, a polynomial
f (x) ∈ k[x] is either irreducible in k[x] or factors as a product of irreducible polynomials in the field k[x]. The polynomial f (x) has to
be of positive degree. Factorization of polynomials play an important
role as it enables solution of polynomial roots as will be seen in the
next section. Indeed, the Groebner basis algorithm presented in Chap.
4 makes use of the factorization of polynomials. In general, computer
algebra systems discussed in Chap. 16 offers possibilities of factoring
polynomials.
3.4 Polynomial Roots
More often than not, the most encountered interaction with polynomials is perhaps the solution of its roots. Finding the roots of polynomials
is essential for most computations that we undertake in practice. As an
example, consider a simple planar ranging case where distances have
been measured from two known stations to an unknown station (see
e.g, Fig. 4.1 on p. 30). In such a case, the measured distances are normally related to the coordinates of the unknown station by multivariate
polynomial equations. If for instance a station P1 , whose coordinates
are {x1 , y1 } is occupied, the distance s1 can be measured to an unknown station P0 . The coordinates {x0 , y0 } of this unknown station
are desired and have to be determined from distance measurements.
The relationship between the measured distance and the coordinates is
given by
p
s1 = (x1 − x0 )2 + (y1 − y0 )2 .
(3.4)
Applying Theorem 3.1, a necessary step to convert (3.4) into polynomial, (3.4) is squared to give a multivariate quadratic polynomial
3.4 Polynomial Roots
s21 = (x1 − x0 )2 + (y1 − y0 )2 .
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(3.5)
Equation (3.5) has two unknowns thus necessitating a second distance
measurement to be taken. Measuring this second distance s2 from station P2 , whose coordinates {x2 , y2 } are known, to the unknown station
P0 leads to a second multivariate quadratic polynomial equation
s22 = (x2 − x0 )2 + (y2 − y0 )2 .
(3.6)
The intersection of the two equations (3.5) and (3.6) results in two
quadratic equations ax20 + bx0 + c = 0 and dy02 + ey0 + f = 0 whose
roots give the desired coordinates x0 , y0 of the unknown station P0 . In
Sect. 4.1, we will expound further on the derivation of these multivariate
quadratic polynomial equations.
In Sect. 3.6, we will discuss the types of polynomials with real coefficients. Suffice to mention at this point that polynomials, as defined in
Definition 3.2 with the coefficients in the field k, has a solution ξ such
that on replacing the variable xα , one obtains
an ξ n + an−1 ξ n−1 + ... + a1 ξ + a0 = 0.
(3.7)
From high school algebra, we learnt that if ξ is a solution of a
polynomial f (x), also called the root of f (x), then (x − ξ) divides
the polynomial f (x). This fact enables the solution of the remaining
roots of the polynomial as we already know. The division of f (x) by
(x − ξ) obeys the division rule discussed in Sect. 3.2.2. In a case where
f (x) = 0 has many solutions (i.e., multiple roots ξ1 , ξ2 , ..., ξm ), then
(x − ξ1 ), (x − ξ2 ), ..., (x − ξm ) all divide f (x) in the field k.
In general, a polynomial of degree n will have n roots that are
either real or complex. If one is operating in the real domain, i.e., the
polynomial coefficients are real, the complex roots normally results in a
pair of conjugate roots. Polynomial coefficients play a significant role in
the determination of the roots. A slight change in the coefficients would
significantly alter the solutions. For ill-conditioned polynomials, such
a change in the coefficients can lead to disastrous results. Methods of
determining polynomial roots have been elaborately presented by [269].
We should point out that for polynomials of degree n in the field of
real numbers R however, the solutions exist only for polynomials up to
degree 4. Above this, Niels Henrick Abel (1802-1829) proved through
his impossibility theorem that the roots are insolvable, while Evariste
Galois (1811-1832) gave a more concrete proof that for every integer n
greater than 4, there can not be a formula for the roots of a general
nt h degree polynomial in terms of coefficients.
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