1
$\begingroup$

Currently, I am reading

Borcea's variance conjectures on the critical points of polynomials.

The following is in this paper:

Let $F(z)=\prod_{j=1}^3 (z-z_j)^{m_j}$ where all the $z_j\in \mathbb{C}$ are distinct and let $w_1,w_2\in \mathbb{C}$ be the zeros of $F'$ that are not zeros of $F.$ Then $$v_1:=\left(\frac{\sqrt{m_1}}{w_1-z_1},\frac{\sqrt{m_2}}{w_1-z_2},\frac{\sqrt{m_3}}{w_1-z_3} \right)\quad \quad and \quad \quad \left(\frac{\sqrt{m_1}}{\overline{w_2}-\overline{z_1}},\frac{\sqrt{m_2}}{\overline{w_2}-\overline{z_2}},\frac{\sqrt{m_3}}{\overline{w_2}-\overline{z_3}} \right)$$ are orthogonal.

It is not clear to me how. Could you explain this relation if you can understand please? Thank you

$\endgroup$

Your Answer

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

Browse other questions tagged or ask your own question.