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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. An example where it does not is given by the isolated singularity of x 2 + y 3 z + z 3 = 0 at the
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]
Because of the order of zeros and poles being defined as a non-negative number n and the symmetry between them, it is often useful to consider a pole of order n as a zero of order –n and a zero of order n as a pole of order –n. In this case a point that is neither a pole nor a zero is viewed as a pole (or zero) of order 0.
Following the terminology in much of the strongly regular graph literature, the larger eigenvalue is called r with multiplicity f and the smaller one is called s with multiplicity g. Since the sum of all the eigenvalues is the trace of the adjacency matrix, which is zero in this case, the respective multiplicities f and g can be calculated:
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).
This is to say that its middle Betti number is equal to the Milnor number and it has homology of a point in dimension less than . For example, a complex plane curve near every singular point z 0 {\displaystyle z_{0}} has its Milnor fiber homotopic to a wedge of μ z 0 ( f ) {\displaystyle \mu _{z_{0}}(f)} circles (Milnor number is a local ...
3.2.1 Zero module. 3.2 ... the intersection multiplicity of several varieties is defined as the length of the ... and its length is less than or equal to its parent ...
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.