Is it assured that if we are making only small changes of coefficients in a polynomial equation
$x^n+a_{n-1}x^{n-1}+... + a_1x +a_0=0$
roots for these equations are changing also by small values? and .. for this case do we have also assured continuity of changes in roots, also in the case when a root is changing from a real value to imaginary one? (i.e. function $\mathbf{x}=f(\mathbf{a})$ - where $\mathbf{x}$ is vector of roots $\mathbf{x}= [x_1,x_2,..x_n]^T$ and $\mathbf{a}$ is vector of coefficients $\mathbf{a}= [a_0,a_1,..a_{n-1}]^T$ - is continuous, smooth, makes no great differential changes in values)
- What theorem does describe this situation ?
Edit (after 9 hours)..
So far there are no answers...
I wonder whether use of Jacobi matrix and Vieta's formulas for the function $\mathbf{a}=f^{-1}(\mathbf{x})$ would not be a good approach.
Below I'm developing this approach:
The equation $x^n+a_{n-1}x^{n-1}+... + a_1x +a_0=0$
can be written as
$\mathbf{x}=f(\mathbf{a})$
where $\mathbf{x}$ is vector of roots and $\mathbf{a}$ is vector of coefficients.
In general case it's not easy to obtain function $\mathbf{f}$, but it is possible from Viete'a formulas
$\begin{cases} x_1 + x_2 + \dots + x_{n-1} + x_n = - {a_{n-1}} \\ (x_1 x_2 + x_1 x_3+\cdots + x_1x_n) + (x_2x_3+x_2x_4+\cdots + x_2x_n)+\cdots + x_{n-1}x_n = {a_{n-2}} \\ {} \quad \vdots \\ x_1 x_2 \dots x_n = (-1)^n {a_0} \end{cases}$
to obtain function $g(\mathbf{x}):\ \mathbf{a}=f^{-1}(\mathbf{x}) = g(\mathbf{x}) $ and from this: $ d\mathbf{a}=J(\mathbf{x})d\mathbf{x}$
For this function we can calculate Jacobi matrix
$\mathbf J = \begin{bmatrix} \dfrac{\partial \mathbf{g}}{\partial x_1} & \cdots & \dfrac{\partial \mathbf{g}}{\partial x_n} \end{bmatrix} = \begin{bmatrix} \dfrac{\partial g_1}{\partial x_1} & \cdots & \dfrac{\partial g_1}{\partial x_n}\\ \vdots & \ddots & \vdots\\ \dfrac{\partial g_n}{\partial x_1} & \cdots & \dfrac{\partial g_n}{\partial x_n} \end{bmatrix}$
$ =\begin{bmatrix} \\ (-1)^nx_2 x_3 \dots x_n & \dots & (-1)^nx_1 x_2 \dots x_{n-1} \\ \vdots & \ddots & \vdots\\ \ x_2 + x_3 + \dots + x_n & \dots & x_1 + x_2 + \dots + x_{n-1} \\ -1 & \cdots & -1 \end{bmatrix}$
Now the problem is to have finite (and possibly small) $\mathbf {J}^{-1}$ $ \dots$ (we have $ d\mathbf{x}=\mathbf{J}^{-1}(\mathbf{x})d\mathbf{a}$)
what is possible only if $det{\mathbf {(J)}}\neq{0}$.
- How to analyze this determinant to get the best results for the starting question?
