This post contains a revised conjecture to a conjecture I posed previously which was shown to be false.

Let $p, q \in \mathbb{C}[t]$ be two monic polynomials of degree $n \ge 1$. For $\alpha \in [0,1]$, let

$$c_\alpha := \alpha p + (1-\alpha)q \in \mathbb{C}[t]$$

Since the zeros of a polynomial vary continuously with respect to its coefficients, following the fundamental theorem of algebra, the locus $$ L(p,q) := \{ z \in \mathbb{C}\mid c_\alpha(z)=0,~\alpha\in[0,1]\} $$ consists of (at most) $n$ continuous paths.

*A priori* determination of the endpoints of the paths is ostensibly difficult.

Notice that if $p$ and $q$ share a simple root $\lambda$, then there is a (degenerate) path from that root to itself.

The picture below contains a typical example: $n=5$." /> generated from the following `MATLAB`

code

```
p=[1 randn(1,5)+i*randn(1,5)]
q=[1 randn(1,5)+i*randn(1,5)]
hold on
scatter(real(roots(p)),imag(roots(p)),'x','r')
scatter(real(roots(q)),imag(roots(q)),'o','b')
for k=0:.01:1
c=k*p+(1-k)*q;
scatter(real(roots(c)),imag(roots(c)),'.','m');
end
```

**Conjecture.** Suppose that $p$, $q \in \mathbb{C}[t]$ have distinct zeros. Let $P = \{ \lambda_1,\dots,\lambda_n\}$ and $Q=\{\mu_1,\dots,\mu_n\}$ denote the zeros of $p$ and $q$, respectively. If there is a disk $D \subset \mathbb{C}$ such that $D \cap P = \mu_i$ and $D \cap Q = \lambda_j$, then there is a path from $\mu_i$ to $\lambda_j$.