nounMany of the same kind; a large number. nounInmathematics, the number of times an object ought to be counted for the sake of regularity. Thus, a zero of a function has a multiplicity of two, if it ought to be regarded as a union of two zeros. This will be shown on a conform...
Zeros of polynomialWe consider multivariate polynomials and investigate how many zeros of multiplicity at least r they can have over a Cartesian product of finite subsets of a field. Here r is any prescribed positive integer and the definition of multiplicity that we use is the one related to ...
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Definition Let L be a linear operator on a finite dimensional vector space, and let λ be an eigenvalue for L. Then the dimension of the eigenspace Eλ is called the geometric multiplicity of λ. Example 8 In Examples 1, 3, and 6 we examined the transpose linear operator on M22 having...
2.1 Boundary Behavior of Herglotz Functions Recall the notion of matrix valued Herglotz functions (often also called Nevanlinna functions). 2.1 Definition An analytic function \(M:{\mathbb {C}}\setminus {\mathbb {R}}\rightarrow {\mathbb {C}}^{n\times n}\) is called a (\(n\times n\)...
Thus, the authors can count the multiplicities of the non-vertical components. In the bivariate case, the authors give a complete algorithm to decompose the system into zeros represented by triangular sets with multiplicities. The authors also analyze the complexity of the algorithm in the bivariate...
,λl) is a weakly decreasing sequence of non-negative integers where only finitely many of the λi are positive. We regard two partitions as the same if they differ only by the number of trailing zeros and call the positive λi the parts of λ. The length is the number of positive ...