Let $p,q$ be monic polynomials in $\mathbb{C}[t]$ and for $\alpha \in [0,1]$, let $c_\alpha := \alpha p + (1-\alpha)q \in \mathbb{C}[t]$. Since the roots 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 $n$ (not necessarily distinct) continuous paths.

*A priori* determination of the endpoints of the paths is ostensiblty diffcult, but experimental evidence suggests the following.

**Conjecture.** Let $P = \{ \lambda_1,\dots,\lambda_n\}$ and $Q=\{\mu_1,\dots,\mu_n\}$ denote the roots of $p$ and $q$, respectively. For $1 \leq i,j\leq n$, let $d(i,j):= |\lambda_i- \mu_j|$. If
$$\text{argmin}_{(i,j)}(d) =\{(k,\ell)\},$$
then there is a path from $\mu_\ell$ to $\lambda_k$.

I am interested in knowing whether the above conjecture is known or if it can be established from what is known from complex analysis and the geometry of polynomials. Note that it is not clear what happens when there is a tie.

Notice that if $p$ and $q$ share a simple root $\lambda$, then there is a (degenerate) path from that root to itself (this corresponds to the case when the distance is zero). The picture below contains a typical example 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
```

**Update:** Per Christian's answer below, the conjecture, as stated above, is not true; however, I am still interested in *a priori* determination of the paths (new question posted).