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The resultant R(x,t) of P and Q with respect to y is a homogeneous polynomial in x and t that has the following property: (,) = with (,) (,) if and only if it exist such that (,,) is a common zero of P and Q (see Resultant § Zeros).
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.
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} .
Assume it is an isolated singularity: in the case of holomorphic mappings it is said that a hypersurface singularity is singular at if its gradient is zero at , and it is said that is an isolated singular point if it is the only singular point in a sufficiently small neighbourhood of . In particular, the multiplicity of the gradient
If y=c is a horizontal asymptote of f(x), then y=c+k is a horizontal asymptote of f(x)+k; If a known function has an asymptote, then the scaling of the function also have an asymptote. If y=ax+b is an asymptote of f(x), then y=cax+cb is an asymptote of cf(x) For example, f(x)=e x-1 +2 has horizontal asymptote y=0+2=2, and no vertical or oblique ...
For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial factorization of x 2 – 4. Factorization is not usually considered meaningful within number systems possessing division , such as the real or complex numbers , since any x {\displaystyle x} can be trivially written as ( x y ) × ( 1 / y ) {\displaystyle ...
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]
Let X be a Riemann surface.Then the intersection number of two closed curves on X has a simple definition in terms of an integral. For every closed curve c on X (i.e., smooth function :), we can associate a differential form of compact support, the Poincaré dual of c, with the property that integrals along c can be calculated by integrals over X: