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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.
If f has a zero of order m at z 0 then for every small enough ρ > 0 and for sufficiently large k ∈ N (depending on ρ), f k has precisely m zeroes in the disk defined by |z − z 0 | < ρ, including multiplicity.
60 = 2 × 2 × 3 × 5, the multiplicity of the prime factor 2 is 2, while the multiplicity of each of the prime factors 3 and 5 is 1. Thus, 60 has four prime factors allowing for multiplicities, but only three distinct prime factors.
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.
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]
The fixed-point index can be thought of as a multiplicity measurement for fixed points. The index can be easily defined in the setting of complex analysis: Let f(z) be a holomorphic mapping on the complex plane, and let z 0 be a fixed point of f. Then the function f(z) − z is holomorphic, and has an isolated zero at z 0.
The resultant of two polynomials depending on a variable x and other variables is a polynomial in the other variables that is in the ideal generated by the two polynomials, and has the following properties: if one of the polynomials is monic in x, every zero (in the other variables) of the resultant may be extended into a common zero of the two ...
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.