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This number of points that have been infinitesimally glued, this local multiplicity of f, is exactly the Milnor number of f. Further contributions [ 2 ] give meaning to Milnor number in terms of dimension of the space of versal deformations , i.e. the Milnor number is the minimal dimension of parameter space of deformations that carry all ...
The graph crosses the x-axis at roots of odd multiplicity and does not cross it at roots of even multiplicity. A non-zero polynomial function is everywhere non-negative if and only if all its roots have even multiplicity and there exists an x 0 {\displaystyle x_{0}} such that f ( x 0 ) > 0 {\displaystyle f(x_{0})>0} .
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]
For example, the rhamphoid cusp y 2 = x 5 has a singularity of order 2 at the origin. After blowing up at its singular point it becomes the ordinary cusp y 2 = x 3, which still has multiplicity 2. It is clear that the singularity has improved, since the degree of defining polynomial has decreased. This does not happen in general.
We define the fixed-point index of f at z 0, denoted i(f, z 0), to be the multiplicity of the zero of the function f(z) − z at the point z 0. In real Euclidean space, the fixed-point index is defined as follows: If x 0 is an isolated fixed point of f, then let g be the function defined by
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.
In this case a point that is neither a pole nor a zero is viewed as a pole (or zero) of order 0. A meromorphic function may have infinitely many zeros and poles. This is the case for the gamma function (see the image in the infobox), which is meromorphic in the whole complex plane, and has a simple pole at every non-positive integer.
Since has zeros inside the disk | | < (because >), it follows from Rouché's theorem that also has the same number of zeros inside the disk. 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).