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: Locus of Roots for <span class=$n=5$." /> 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
for k=0:.01:1

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 $\lambda_j$.

  • 2
    $\begingroup$ Doesn't the same counterexample work here? Consider the disk centered at 0 with radius 0.2 $\endgroup$ – Fan Zheng Jan 31 '18 at 19:56
  • $\begingroup$ In the expression $L(p,q) := \{ z \in \mathbb{C}\mid c_\alpha(z)=0,~\alpha\in[0,1]\}$, it is beneficial to allow more general values of $\alpha$. First allow all $\alpha\in\mathbb R$. Then allow $\alpha\in\mathbb C$ with $\mathrm{im}(\alpha)\in [0,1]$, etc. $\endgroup$ – André Henriques Jan 31 '18 at 20:49
  • $\begingroup$ @AndréHenriques: I don't disagree with your suggestion — the convex parameter arises from my research. $\endgroup$ – Pietro Paparella Jan 31 '18 at 21:10

Your new conjecture is a stronger version of your previous conjecture.

Suppose your new conjecture was true, and suppose that $\mu_i, \lambda_j$ are roots of $p$ and $q$ respectively such that the distance $d(\mu_i,\lambda_j)$ is minimal. Your previous conjecture was that there is a path from $\mu_i$ to $\lambda_j$.

The disc centred at $\frac{\mu_i+\lambda_j}{2}$ with radius $\frac{1}{2}d(\mu_i,\lambda_j)$ contains only the two roots $\mu_i, \lambda_j$. So your new conjecture would imply that there is a path from $\mu_i$ to $\lambda_j$, that is that your previous conjecture was true.

Perhaps, as André Henriques suggests, it would be helpful to first consider more general values of $\alpha$ in order to understand the particular case $\alpha\in[0,1]$.

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