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For univariate polynomials with complex coefficients, factorization can easily be reduced to numerical computation of polynomial roots and multiplicities. In the multivariate case, a random infinitesimal perturbation of the coefficients produces with probability one an irreducible polynomial, even when starting from a polynomial with many factors.
For polynomials with real or complex coefficients, it is not possible to express a lower bound of the root separation in terms of the degree and the absolute values of the coefficients only, because a small change on a single coefficient transforms a polynomial with multiple roots into a square-free polynomial with a small root separation, and ...
So, except for very low degrees, root finding of polynomials consists of finding approximations of the roots. By the fundamental theorem of algebra, a polynomial of degree n has exactly n real or complex roots counting multiplicities. It follows that the problem of root finding for polynomials may be split in three different subproblems;
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 solutions of the system are in one-to-one correspondence with the roots of h and the multiplicity of each root of h equals the multiplicity of the corresponding solution. The solutions of the system are obtained by substituting the roots of h in the other equations. If h does not have any multiple root then g 0 is the derivative of h.
Polynomials of degree one, two or three are respectively linear polynomials, quadratic polynomials and cubic polynomials. [8] For higher degrees, the specific names are not commonly used, although quartic polynomial (for degree four) and quintic polynomial (for degree five) are sometimes used. The names for the degrees may be applied to the ...
3 Cyclotomic polynomials over a finite field and over the p-adic integers. 4 Polynomial values. 5 Applications. ... If p is a prime divisor of multiplicity m in n, ...
If the multiplicity m of the root is finite then g(x) = f(x) / f ′ (x) will have a root at the same location with multiplicity 1. Applying Newton's method to find the root of g ( x ) recovers quadratic convergence in many cases although it generally involves the second derivative of f ( x ) .