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Wednesday, 20 November 2013

Root of Polynomials

 

Definition :-   Let  F  be any field and  p(x)  be any polynomial in   F[x]   . Then an element    a    lying in some extension field of    F   is called a root of   p(x)   if   p(a) = 0   .

from the theorems of polynomials we know that the element lying in an extension    K    of    F    which were algebraic over   F   i.e. which satisfies a polynomial

         p(x)  is element of   F[x]    i.e.    p(a) = 0

Here in this article we shall aim at finding an extension field    K   of   F    in which a given polynomial

                              p(x)   is an element of   F[x]   has a root .

The statistical properties of the roots of a random polynomial have been the subject of several studies. Let
 p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0
be a random polynomial. If the coefficients ai are independently and identically distributed with a mean of zero, the real roots are mostly located near ±1. The complex roots can be shown to be on or close to the unit circle.
If the coefficients are Gaussian distributed with a mean of zero and variance of σ then the mean density of real roots is given by the Kac formula
 m( x ) = \frac { \sqrt{ A( x ) C( x ) - B( x )^2 }} {\pi A( x )}
where
 \begin{align}
A( x ) &= \sigma \sum { x^{ 2i } } = \sigma \frac{ x^{ 2n } - 1 } { x - 1 },  \\
B( x ) &= \frac{ 1 } { 2 } \frac{ d } { dt } A( x ), \\
C( x ) &= \frac{ 1 } { 4 } \frac{ d^2 } { dt^2 } A( x ) + \frac{ 1 } { 4x } \frac{ d } { dt } A( x ). 
\end{align}
When the coefficients are Gaussian distributed with a non zero mean and variance of σ, a similar but more complex formula is known .