\( \def\RR{{\mathbb{R}}} \) \( \def\ZZ{{\mathbb{Z}}} \)

Nice Polynomials of degree 3

Here follows an example inspired by some great mathematical work of Jean-François Burnol.

All graphics are setup with PSTricks. All animations are generated with the animate package of Alexander Grahn and finally converted to svg format. The calculations within the animations are mostly done with xintexpr (part of the mighty xint bundle) of Jean-François Burnol.

The dance of the real roots of a polynomial of degree 3

Short mathematical background

Consider a cubic equation with real coefficients $P=0$ with three distinct roots. These are the abscissas of the vertices of an equilateral triangle whose radius is equal to the horizontal distance between the local extrema of the graph of the given cubic polynomial. The animation shows this fact in varying the second term of the equation $P = u$, which corresponds to the intersection of the graph with the horizontal line $y=u$.

The dance of the rational roots of a polynomial of degree 3

A much more complicated thing to find only rational roots solved and animated by Jean-François Burnol. All calculations within this animation are done with xintexpr of Jean-François Burnol.

Note: This animation can't be fluently due to only rational roots.

The dance of the integer roots of a polynomial of degree 3

Even much more complicated thing to find only integer roots solved by Jean-François Burnol.

As an example we chose the cubic: $P(X)=(X-10)(X+1)(X+9)=X^3-91X-90$

The roots of the first derivative are not integers.

Note: This animation can't be fluently due to only integer roots.

The dance of the integer roots of a $\ZZ$-nice polynomial of degree 3

Even much much more complicated thing to find only integer roots and integer roots of the first derivative solved by Jean-François Burnol.

As an example we chose the $\ZZ$-nice cubic: $P(X)=X(X-9)(X-24)=X^3-33X^2+216X$

This polynomial gives the minimal distance of the abscissas of its local extrema which is 14.

Note: This animation can't be fluently due to only integer roots.

Symmetry dancing ...

Documentation

PDF Nice cubic polynomials: symmetry and arithmetic of the Lagrange resolvent
by Jean-François Burnol and Jürgen Gilg (in English)

PDF La voie-sans-effort™ vers le cubisme
by Jean-François Burnol (in French)

WWW Polynômes cubiques plaisants
Excerpts of the documentation by Manuel Luque (in French)

WWW Polynômes cubiques plaisants (2)
Excerpts of the documentation by Manuel Luque (in French)

The $\TeX$ source files can be accessed at

WWW Polynômes plaisants (figures et animation avec PSTricks, calculs avec xint, en particulier les fractions)
Many thanks to Jean-Michel Sarlat