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Preliminaries
Published in Ronald B. Guenther, John W. Lee, Sturm-Liouville Problems, 2018
Ronald B. Guenther, John W. Lee
The fundamental theorem of algebra states that a polynomial of degree n has exactly n zeros when each zero is counted to its multiplicity. A zero c of a polynomial p(x) has multiplicity m if p(c)=⋯=p(m−1)(c)=0 and p(m)(c)≠0. A zero is simple if its multiplicity is 1 and is a double zero if its multiplicity is 2. A polynomial changes it sign at a simple zero and maintains a fixed sign near a double zero. Multiplicity in this sense does not apply to a ϕ-polynomial, unless it is sufficiently differentiable. But the zeros of a ϕ-polynomial can be counted in a way that distinguishes between zeros where a sign change occurs and those where no sign change occurs.
Variables, functions and mappings
Published in Alan Jeffrey, Mathematics, 2004
The number of times a root of an equation is repeated is called the multiplicity of the root. Thus if P7(x) is expressible in factorized form as P7(x)=(x−ζ1)2(x−ζ2)(x−ζ3)4, it follows that ζ1 is a root of multiplicity 2, ζ2 is a single root (multiplicity 1) and ζ3 is a root of multiplicity 4.
Polynomials, Roots, the Principle of the Argument, and Nyquist Stability
Published in A. David Wunsch, ® Companion to Complex Variables, 2018
To explore the fundamental theorem in MATLAB, it is easiest to use the function roots. The reader may wish to review the documentation for this function using the MATLAB help. Briefly, to find the roots of a polynomial expression like Equation 5.1, you must begin by creating a row vector whose elements are the coefficients of the various powers of z listed in descending order of the associated powers. The first element would be the number an. If any power of z that is less than n does not appear in the polynomial, then you must enter a zero for that coefficient in the vector. If a polynomial has a root of multiplicity n, the output of roots will show that root n times. To find the roots of the polynomial z4 + z3 + 2z + 3, we proceed as follows in MATLAB: a = [1 1 0 2 3]>> roots(a)ans = 0.7140 + 1.1370i0.7140 − 1.1370i−1.2140 + 0.4366i−1.2140 − 0.4366i
Multidimensional realisation theory and polynomial system solving
Published in International Journal of Control, 2018
Philippe Dreesen, Kim Batselier, Bart De Moor
In the one-dimensional case, the Fundamental Theorem of Algebra states that a univariate degree d polynomial f(x) has exactly d roots in the field of complex numbers. When several of these roots coincide, we say that they occur with multiplicity. This happens if f(x) has a horizontal tangent at the position of a multiple root. The multidimensional counterpart of the Fundamental Theorem of Algebra is called Bezout's theorem (see Cox et al. 2005, p. 97 and Shafarevich 2013, p. 246). This theorem states that a set of Equations (18) that describes a zero-dimensional solution set has exactly m = ∏idi solutions in the projective space, counted with multiplicity.