# 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 general, this type of problem can be very difficult. To see some of the difficulties, check out Wilkinson's Polyonmial, specifically, the conditioning and stability parts.

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.