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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 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 prime factorization, the multiplicity of a prime factor is its -adic valuation.For example, the prime factorization of the integer 60 is . 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.
The number of the common zeros of the polynomials in a Gröbner basis is strongly related to the number of monomials that are irreducibles by the basis. Namely, the number of common zeros is infinite if and only if the same is true for the irreducible monomials; if the two numbers are finite, the number of irreducible monomials equals the ...
This shows that the number of common zeros is at most the degree of the resultant, that is at most the product of the degrees of P and Q. With some technicalities, this proof may be extended to show that, counting multiplicities and zeros at infinity, the number of zeros is exactly the product of the degrees.
Theorem — The number of strictly positive roots (counting multiplicity) of is equal to the number of sign changes in the coefficients of , minus a nonnegative even number. If b 0 > 0 {\displaystyle b_{0}>0} , then we can divide the polynomial by x b 0 {\displaystyle x^{b_{0}}} , which would not change its number of strictly positive roots.
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.
Every real polynomial of odd degree has an odd number of real roots (counting multiplicities); likewise, a real polynomial of even degree must have an even number of real roots. Consequently, real odd polynomials must have at least one real root (because the smallest odd whole number is 1), whereas even polynomials may have none.