Lipschitz-like behaviour of quartic polynomials

Question

I have observed the following phenomenon:

Let the biquadratic $q(x)=x^4-Ax^2+B$ have four real roots and perturb it by a linear factor $p(x)=q(x)+mx$, so that $m$ not too large with respect to $A,B$.

Then the roots of $p(x)$ (assuming they are real) are very close to the roots of $q(x)$.

Example: $q(x)=x^4-15x^2+20$ has roots $\pm 1.22,\pm 3.68$. The perturbation $q(x)-4x$ has roots $-3.5,-1.4,1.065,3.83$ (I rounded to 1-2 digits for clarity).

The difference between corresponding roots of $q(x)$ and $r(x)$ does not exceed 0.185 in this case, which I consider quite close (good enough for my purposes).

Is there an explanation for this phenomenon?

I found two relevant papers, however one seems to have uncspecified constants $C$ and another goes into deep theory. I am looking for a simple, usable, result or at least for a simple explanation, if there is one.

Answer 1

You should look at that source of all wisdom: Kato's Perturbation theory of linear operators, which, in the first chapter, discusses the perturbation of eigenvalues of a matrix (before plunging into infinite dimensions in subsequent sections). Apply that discussion to the companion matrix of your polynomial.

Of course, for quartics, there is an explicit formula in radicals, so you can write down the perturbation as explicitly as you like using that.

Answer 2

To expand slightly on Alexandre's comment: polynomial roots are stable under small perturbations. Suppose $p(z,\lambda) = \sum_{j=0}^n a_j(\lambda) z^j$ is a polynomial with coefficients that are continuous functions of parameter $\lambda$, and $\Gamma$ is a simple closed contour in the complex plane. Then by the Argument Principle the number of roots (counted by multiplicity) inside $\Gamma$ is constant as long as there are no roots on $\Gamma$. Moreover, in the case where there is one simple root inside $\Gamma$ the location of that root can be computed as

$$ \dfrac{1}{2\pi i} \oint_\Gamma \dfrac{z\; p_z(z,\lambda)}{p(z,\lambda)}\; dz$$ (where $p_z$ is partial derivative wrt $z$).

By appropriate estimates you can use this to control how the location of the root depends on $\lambda$.

Another way to study the dependence of the roots on the parameter is by the differential equation

$$ \dfrac{dz}{d\lambda} = - \dfrac{p_\lambda(z,\lambda)}{p_z(z,\lambda))}$$

EDIT: In the case at hand, with $p(z,m) = z^4 - 15 z^2 + m z + 20$, the differential equation is

$$ \dfrac{dz}{dm} = \dfrac{-z}{4 z^3 -30 z + m } $$

At $m=0$ and $z$ the root near $+1.22$, for example, $dz/dm \approx 0.0415$. It stays positive and $< 0.06$ for $-\infty < m < 6.178$ approximately, only getting large as you approach $m \approx 13.8836$ where the discriminant of $P$ hits $0$ and this root collides with another. Here is a plot of the real roots as they depend on $m$.