# Polynomial roots conditions vary with coefficients

Polynomial equation $\sum_{i=0}^4 p_i x^i=0$ have the following root conditions:

1) $a_0 \pm b_0 i, a_1 \pm b_1i$
2) $a_0 \pm b_0 i, a_1, a_2$
3) $a_0, a_1, a_2 \pm b_2i$
4) $a_0, a_1, a_2, a_3$

I'm interested in how the root conditions swap with infinitesimal changes in coefficients $p_i$. For instance, condition 1) may change to condition 2) for an infinitesimal change in $p_i$, however, it is unlikely to jump directly to condition 4).

Is there some rules about the permutation of root condition (1-4) with respect to infinitesimal changes of coefficients? The permutations become more tricky when high degree polynomials.

In the case you describe, if you have real coefficients, then starting with $4$ distinct real roots, the roots will move continuously as the coefficients vary and remain real until two roots coincide, in which case, the roots can split off to a pair of complex conjugates.
Also, your cases $2$ and $3$ are the same since there is no ordering among the roots of a polynomial.