1
$\begingroup$

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
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 $\lambda_j$.

$\endgroup$
  • 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
1
$\begingroup$

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 $\alpha\in[0,1]$.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.