2

A complex cubic polynomial arranged as $f(z)=z^3-3az^2+3b^2z-c^3$ with coefficients $a,b,c\in\mathbb{C}$ can be represented as the product of (unknown) factors $f(z)=(z-p)(z-q)(z-r)=z^3-3\left(\frac{p+q+r}{3}\right)z^2+3\left(\sqrt \frac{pq+qr+rp}{3}\right)^2z-\left(\sqrt[3]{pqr}\right)^3$ with roots $p,q,r\in\mathbb{C}$.

Thus we can read the centroid $a=\frac{p+q+r}{3}$ of the Steiner ellipse of the triangle $p,q,r\in\mathbb{C}$ right from the coefficients (e.g. reasoning with Marden’s theorem).

Do the coefficients $b=\sqrt \frac{pq+qr+rp}{3}$ and $c=\sqrt[3]{pqr}$ have similar nice intuitive geometric interpretation, e.g. describe characteristic points or features of the triangle or its Steiner ellipse?

dontpanic
  • 91
  • 2
    In other words, do the elementary symmetric polynomials in $3$ (or $n$) complex variables have a geometric interpretation? – Justpassingby Jan 07 '16 at 13:07
  • they don't have a geometric because they don't commute with translations. If you add $1$ to $p,q,r$ then the new $b$ and $c$ aren't the old ones plus $1$. – mercio Jan 07 '16 at 13:12
  • @Justpassingby The polynomial is only in one complex variable $z$. – dontpanic Jan 08 '16 at 06:43
  • @mercio that's not the question here, the roots $p,q,r$ will naturally change as we change the given $a,b,c$. However, the roots are unknown, I'm just adding the factorization to provide the intuition for the underlying complex triangle. Still, given $a,b,c$ alone we can say something about the triangle $p,q,r$ right from the original polynom coefficients, e.g. $a$ is the centroid or arithmetic mean, and $c$ is the geometric mean etc. Any intuition about $b$? – dontpanic Jan 08 '16 at 06:43

2 Answers2

3

Vieta's Formulae

\begin{align} f(z) &= z^3-3az^2+3b^2z-c^3 \\ &= (z-p)(z-q)(z-r) \\ a &= \frac{p+q+r}{3} \\ b^2 &= \frac{pq+qr+rp}{3} \\ c^3 &= pqr \end{align}

Foci of Steiner ellipse

\begin{align} f'(\lambda) &=0 \\ \lambda &= a\pm \sqrt{a^2-b^2} \\ &= \frac{p+q+r \pm \sqrt{(p+q+r)^2-3(pq+qr+rp)}}{3} \end{align}

Resolvents \begin{align} u &= \sqrt[3]{\frac{c^3+3ab^2-2a^3}{2}+ \sqrt{(b^2-a^2)^3+ \left( \frac{c^3+3ab^2-2a^3}{2} \right)^2}} \\ v &= \sqrt[3]{\frac{c^3+3ab^2-2a^3}{2}- \sqrt{(b^2-a^2)^3+ \left( \frac{c^3+3ab^2-2a^3}{2} \right)^2}} \\ 0 &= f(a+u\, \omega^n+v\, \omega^{2n}) \tag{$\omega=e^{2\pi i/3}$} \end{align}

  • Centroid $$a=\frac{p+q+r}{3}$$

  • Linear eccentricity of Steiner ellipse $$\sqrt{|a^2-b^2|}=\sqrt{|uv|}$$

  • Semi-major axis of Steiner ellipse $$\frac{|u|+|v|}{2}$$

  • Semi-minor axis of Steiner ellipse $$\frac{||u|-|v||}{2}$$

Ng Chung Tak
  • 18,990
-1

The geometric meaning of roots is that they are points on coordinate axis where the derivative or slope of a given graph becomes $0$ or the place where graph touches axes . Hope i have interpreted it correctly.

Archis Welankar
  • 15,910
  • 1
    Thanks, but I'm not looking for meaning of roots (which are simply points in the complex plane), but meaning of the coefficients of the complex polynom. These coefficients are however functions of the roots. Looking closer at these functions, can we interpret them geometrically? E.g. arithmetic mean of roots, geometric mean of roots, eccentricity of Steiner ellipse, etc? – dontpanic Jan 08 '16 at 06:47