Algebra

Invariants of curves of second order

  • version 1.0, 2013-09-13 : pdf
    MSC Primary 14H50

Conics occur as orbits of massive bodies in a gravitational field,
hence they are important in Astronomy.
The orbital elements are fundamental invariants of the curve.
Determining the invariants of curves of second order
a11x2 +2a12xy + a22y2 + 213x +2a23y + a33 = 0
from the coefficients aik of the equation is a well known task.
We derive formulas for the invariants following an elementary high school
course developed by Freudenthal and Heinrich 50 years ago.

Note on adjoint functors

  • version 1.0, 2011-04-25 : pdf
    MSC Primary 18A40, Secondary 18A30

Basic definition of categories, functors, morphism of functors, adjunction of functors.

Plane curves of second order and their invariants

Neue Behandlung der Kurven zweiter Ordnung durch Invarianten
von Ernst Freudenthal und Werner Heinrich

  • Version 1.0, rev. 504, 2015-03-04 : pdf (deutsch !)
    first published: 2010-04-21
    MSC Primary 14H50
  • Version 1.0, rev. 504, 2015-03-04 : pdf (deutsch !) Lösungsheft zu allen Aufgaben
    first published: 2011-11-30

In grateful souvenir to Werner Heinrich, who introduced me to Mathematics.

In this course the plane curves of second order are treated by their invariants. The approach is basically algebraic, although several geometric constructions (focal generation, pencils of curves, Pascal line configuration) are given as well.

The course contains several detailed examples and about 250 exercises. In this TeX version I have generated some 20 figures with METAPOST.

The recent revision fixed several typos. The solution booklet to all the exercises is published as well.

Cardano’sche Formeln

  • version 1.1, rev. 491, 2015-03-04 : pdf (deutsch !)
    first published: 2010-04-05
    MSC Primary 12E05, Secondary 12F10

Für meinen Bruder Klaus.

A short note on Cardano‘s and Ferrari‘s equations for solving cubic and biquadratic equations.

Eine kurze Note über die Formeln von Cardano und Ferrari zur Lösung von kubischen und biquadratischen Gleichungen durch Ziehen von quadratischen und kubischen Wurzeln aus den Koeffizienten.

In der aktualisierten Version 1.1 habe ich das Vorwort korrigiert, nachdem ich die Literaturangaben überprüft habe (in Jantzen-Schwermer wird die Cardano Formel nur ohne Beweis angegeben, zu Ferrari steht dort nichts).

A note on Separability

  • version 1.1, 515, 2015-03-04 : pdf
    first published: 2004-01-06
    MSC Primary 12F10, Secondary 12F15

This note gathers key notions and proofs around separability with only modest prerequisites, covered by a first course in algebra. This revision contains only minor rearrangements vs. the original version from January 2004.