# Locus of roots of all convex combinations of two monic polynomials, II

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 $$\lambda_j$$.

• Doesn't the same counterexample work here? Consider the disk centered at 0 with radius 0.2 – Fan Zheng Jan 31 '18 at 19:56
• 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. – André Henriques Jan 31 '18 at 20:49
• @AndréHenriques: I don't disagree with your suggestion — the convex parameter arises from my research. – 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 $\alpha\in[0,1]$.