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The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. [3]
In algebraic graph theory it equals the multiplicity of 0 as an eigenvalue of the Laplacian matrix of a finite graph. [14] It is also the index of the first nonzero coefficient of the chromatic polynomial of the graph, and the chromatic polynomial of the whole graph can be obtained as the product of the polynomials of its components. [15]
The intersection number arises in the study of fixed points, which can be cleverly defined as intersections of function graphs with a diagonals. Calculating the intersection numbers at the fixed points counts the fixed points with multiplicity, and leads to the Lefschetz fixed-point theorem in quantitative form.
If the graph has n vertices and m edges, then: In the matrix theory of graphs, the nullity of the graph is the nullity of the adjacency matrix A of the graph. The nullity of A is given by n − r where r is the rank of the adjacency matrix. This nullity equals the multiplicity of the eigenvalue 0 in the spectrum of the adjacency matrix. See ...
In numerical analysis, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function f is a number x such that f ( x ) = 0 . As, generally, the zeros of a function cannot be computed exactly nor expressed in closed form , root-finding algorithms provide approximations to zeros.
One advantage of this proof over the others is that it shows not only that a polynomial must have a zero but the number of its zeros is equal to its degree (counting, as usual, multiplicity). Another use of Rouché's theorem is to prove the open mapping theorem for analytic functions. We refer to the article for the proof.
where the zero and one entries of are treated as numerical, rather than logical as for simple graphs, values, explaining the difference in the results - for simple graphs, the symmetrized graph still needs to be simple with its symmetrized adjacency matrix having only logical, not numerical values, e.g., the logical sum is 1 v 1 = 1, while the ...
The concept of multiplicity is fundamental for Bézout's theorem, as it allows having an equality instead of a much weaker inequality. Intuitively, the multiplicity of a common zero of several polynomials is the number of zeros into which the common zero can split when the coefficients are slightly changed.