Basics of Polynomial Theory


 Prudence Marshall
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1 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 x α 1 1 xα xαn n, (α 1,..., α n ) Z n + in the variables x 1,..., x n. In Definition 3.1 above, the set Z n + 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 threedimensional resection problem given as (see e.g., (11.44) on p. 180) x a 12x 1 x 2 + x a o = 0 x b 23x 2 x 3 + x b o = 0 x c 31x 3 x 1 + x c o = 0 where x 1 R +, x 2 R +, x 3 R +.
2 18 3 Basics of Polynomial Theory The variables {x 1, x 2, x 3 } are unknowns while the other terms are known constants. The products of variables { x 2 1, x 1x 2, x 2 2, x 2x 3, x 2 3, x 3x 1 } are monomials in {x 1, x 2, x 3 }. Summation of monomials form polynomials defined as Definition 3.2 (Polynomial). A polynomial f k [x 1,...,x n ] in variables x 1,...,x n with coefficients in the field k is a finite linear combination of monomials with pairwise different terms expressed as f = α a α x α, a α k, x α = (x α 1,..., x αn ), α = (α 1,..., α n ), (3.1) where a α are coefficients in the field k, e.g., R or C and x α the monomials. Example 3.2 (Polynomials). Equations x a 12x 1 x 2 + x a o = 0 x b 23x 2 x 3 + x b o = 0 x c 31x 3 x 1 + x c o = 0, in Example 3.1 are multivariate polynomials. The first expression is a multivariate polynomial in two variables {x 1, x 2 } and a linear combination of monomials { x 2 1, x 1x 2, x 2 2}. The second expression is a multivariate polynomial in two variables {x 2, x 3 } and a linear combination of the monomials { x 2 2, x 2x 3, x 2 3}, while the third expression is a multivariate polynomial in two variables {x 3, x 1 } and a linear combination of the monomials { x 2 3, x 3x 1, x 2 1}. 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 9w 2 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 9 Q, then the solution w = ± 1 3 exist.
3 3.2 Polynomial Rings 19 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 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 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 a i R to the variable and obtaining summation over finite number of distinct integers. Thus f(x) = c α x α, c α R, α 0 α is said to be a univariate polynomial over R. If two polynomials are given such that f 1 (x) = c i x i and f 2 (x) = d j x j, then two binary i j operations addition and multiplication can be defined on these polynomials such that: (a) Addition: f 1 (x) + f 2 (x) = e k x k, e k = c k + d k, e k R k (b) Multiplication: f 1 (x).f 2 (x) = g k x k, g k = c i d j, g k R. 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 c i 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 x 1,...,x n, 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 x i over k expressed as P := k [x 1,..., x n ]. Its elements are polynomials
4 20 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 x a 12x 1 x 2 + x a o = 0 x b 23x 2 x 3 + x b o = 0 x c 31x 3 x 1 + x c o = 0 of Example 3.1 are said to be polynomials in three variables [x 1, x 2, x 3 ] forming elements of the polynomial ring P over the field of real numbers R expressed as P := R [x 1, x 2, x 3 ]. 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 f i k{ξ 1,..., ξ m } expressed as f 1 k{ξ 1,..., ξ m } f 2 k{ξ 1,..., ξ m }. (3.2).. f n k{ξ 1,..., ξ m }, with the order considered as l, we write the objective function to be minimized as
5 3.2 Polynomial Rings 21 f 2 = f f 2 n f i k{ξ 1,..., ξ m }, (3.3) and obtain the partial derivatives (first derivatives of 3.3) with respect to the unknown variables {ξ 1,..., ξ m }. The order of (3.3) which is l 2 then reduces to (2l 1) upon differentiating the objective function with respect to the variables ξ 1,..., ξ m. Thus resulting in m normal equations of the polynomial order (2l 1). Example 3.5 (Pseudoranging problem). For pseudoranging or distance equations, the order of the polynomials in the algebraic observational equations is l = 2. If we take the pseudoranges squared or distances squared, a necessary procedure inorder 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 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).
6 22 3 Basics of Polynomial Theory For univariate polynomials, as in Definition 3.4, the Euclidean algorithm employs operations addition and multiplication to factor polynomials inorder to reduce them to satisfy the definition of division algorithm. 3.3 Factoring Polynomials Inorder 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 P 1, whose coordinates are {x 1, y 1 } is occupied, the distance s 1 can be measured to an unknown station P 0. The coordinates {x 0, y 0 } 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 s 1 = (x 1 x 0 ) 2 + (y 1 y 0 ) 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
7 3.4 Polynomial Roots 23 s 2 1 = (x 1 x 0 ) 2 + (y 1 y 0 ) 2. (3.5) Equation (3.5) has two unknowns thus necessitating a second distance measurement to be taken. Measuring this second distance s 2 from station P 2, whose coordinates {x 2, y 2 } are known, to the unknown station P 0 leads to a second multivariate quadratic polynomial equation s 2 2 = (x 2 x 0 ) 2 + (y 2 y 0 ) 2. (3.6) The intersection of the two equations (3.5) and (3.6) results in two quadratic equations ax bx 0 + c = 0 and dy ey 0 + f = 0 whose roots give the desired coordinates x 0, y 0 of the unknown station P 0. 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 x α, one obtains a n ξ n + a n 1 ξ n a 1 ξ + a 0 = 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 In a case where f(x) = 0 has many solutions (i.e., multiple roots ξ 1, ξ 2,..., ξ m ), then (x ξ 1 ), (x ξ 2 ),...,(x ξ m ) 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 illconditioned 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 ( ) proved through his impossibility theorem that the roots are insolvable, while Evariste Galois ( ) 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 n t h degree polynomial in terms of coefficients.
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