Prof. Dr. Martin Kreuzer SS 2001 Fachbereich Mathematik Universität Bayreuth

Übungen zur Vorlesung

Praktische Anwendungen der Computeralgebra 2

Blatt 4

Abgabetermin: Do, 26.7.2001

Aufgabe 16: (Hyperebenen im projektive Raum) Sei K ein Körper und H ⊆ ℙ_{K}^{n} eine Hyperebene.

a) Finden Sie ein Tupel (a_{0}, …, a_{n}) ∈ K^{n+1} \ {0}, so dass H aus allen Punkten (p_{0} : …: p_{n}) besteht für die a_{0}p_{0} + ⋯ + a_{n}p_{n} = 0 ist. (Zeigen Sie zuerst, dass diese Bedingung wohldefiniert ist.)

b) Beweisen Sie, dass zwei Tupel (a_{0}, …, a_{n}) und (b_{0}, …, b_{n}) genau dann auf die in a) beschriebene Weise die gleiche Hyperebene H ⊆ ℙ_{K}^{n} definieren, wenn es ein Element λ ∈ K \ {0} gibt mit (a_{0}, …, a_{n}) = λ · (b_{0}, …, b_{n}).

c) Zeigen Sie, dass sich zwei Geraden in ℙ_{K}^{2} immer in genau einem Punkt schneiden.

Aufgabe 17: (Der duale projektive Raum) Sei K ein Körper und sei Hyp(ℙ_{K}^{n}) die Menge der Hyperebenen in ℙ_{K}^{n}.

a) Zeigen Sie mit Hilfe von Aufgabe 16, dass es eine wohldefinierte Abbildung η : Hyp(ℙ_{K}^{n}) ⟶ ℙ_{K}^{n} gibt, die H auf (a_{0}, …, a_{n}) abbildet.

b) Beweisen Sie, dass die Abbildung η bijektiv ist. Daher können wir die Menge Hyp(ℙ_{K}^{n}) als n-dimensionalen projektiven Raum über K betrachten. Wir nennen ihn den dualen projektiven Raum und schreiben (ℙ_{K}^{n})ˇ .

c) Sei S die Menge aller Punkte in (ℙ_{K}^{n})ˇ , die zu Hyperebenen durch den Punkt (0 : …: 0 : 1) gehören. Zeigen Sie, dass S eine Hyperebene in (ℙ_{K}^{n})ˇ ist.

d) Nun sei p eine Primzahl und K der endliche Körper K = 𝔽_{p}. Schreiben Sie eine CoCoA-Funktion Hyperplanes(…), die ausgehend von einem Punkt (p_{0} : …: p_{n}) in ℙ_{K}^{n} die Liste aller Tupel (a_{0}, …, a_{n}) ∈ K^{n+1} berechnet, die zu den Hyperebenen durch diesen Punkt gehören. Wenden Sie Ihre Funktion im Fall p = 3 auf die folgenden Punkte an.

1) P = (0 : 0 : 1) ∈ ℙ_{K}^{2}

2) P = (0 : 1 : 1) ∈ ℙ_{K}^{2}

3) P = (1 : 1 : 1) ∈ ℙ_{K}^{2}

Aufgabe 18: (Grassmannsche Varietäten) Sei K ein Körper und sei Lin(ℙ_{K}^{3}) die Menge der Geraden in ℙ_{K}^{3}.

a) Erklären Sie, wie man Lin(ℙ_{K}^{3}) mit der Menge aller 2-dimensionalen K-Unter-vektorräume von K^{4} identifizieren kann.

b) Nun definieren wir eine Abbildung φ : Lin(ℙ_{K}^{3}) ⟶ ℙ_{K}^{5} wie folgt.

Repräsentiere den zu einer Geraden L ⊆ ℙ_{K}^{3} gehörigen 2-dimensionalen Vektorraum V ⊆ K^{4} durch eine Matrix

V = ( \begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \\ a_{41} & a_{42} \end{matrix} )

und bilde das 6-Tupel der 2 × 2-Minoren (d_{12}, d_{13}, d_{14}, d_{23}, d_{24}, d_{34}), wobei d_{ij} zu den Zeilen i und j gehöre. Nun setze φ(L) = (d_{12} : d_{13} : d_{14} : d_{23} : d_{24} : d_{34}). Beweisen Sie, dass die Abbildung φ wohldefiniert und injektiv ist. Ihr Bild heisst die Grassmannsche Varietät der Geraden in ℙ_{K}^{3} und wird mit Grass_{1}(ℙ_{K}^{3}) bezeichnet. Hinweis: Zeigen Sie, das man o.E. d_{12} ≠ 0 annehmen kann sowie a_{ij} = δ_{ij} für i, j = 1, 2.

c) Nun wollen wir noch zeigen, dass Grass_{1}(ℙ_{K}^{3}) eine projektive Varietät in ℙ_{K}^{5} ist. Dazu beweisen Sie zunächst, dass die Minoren d_{ij} der folgenden Gleichung genügen:

d_{12}d_{34} − d_{13}d_{24} + d_{14}d_{23} = 0

d) Zeigen Sie mit Hilfe von CoCoA, dass die Gleichung in c) die einzige ist, die unter diesen Minoren gilt. Definieren Sie dazu ein passendes Ideal I im Polynomring

P = K[d_{12}, d_{13}, d_{14}, d_{23}, d_{24}, d_{34}, a_{11}, a_{12}, a_{21}, a_{22}, a_{31}, a_{32}, a_{41}, a_{42}]

und berechnen Sie das Eliminationsideal I ∩ K[d_{12}, …, d_{34}]. Folgern Sie, dass Grass_{1}(ℙ_{K}^{3}) die von einer quadratischen Gleichung definierte Hyperfläche in ℙ_{K}^{5} ist.

Aufgabe 19: (Graphenfärbungen) Sei ein Graph gegeben durch die Nummern {1, …, n} seiner Ecken und die Paare in {1, …, n}^{2}, die seinen Kanten entsprechen.

a) Schreiben Sie eine CoCoA-Funktion Colouring(…), die die Liste der Kanten nimmt und ein Ideal I in P = 𝔽_{3}[x_{1}, …, x_{n}] berechnet, dessen Nullstellen genau die 3-Färbungen des Graphen sind. Dabei sei o.E. die Ecke Nr. 1 mit “0” gefärbt und Nr. 2 mit “1”.

b) Wenden Sie Ihre Funktion Colouring(…) auf den folgenden Graph an:

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Bestimmen Sie mit Hilfe von CoCoA die reduzierte Lex-Gröbner Basis dieses Ideals. Hat der Graph eine 3-Färbung? Wenn ja, wieviele verschiedene 3-Färbungen gibt es?

c) Betrachten Sie nun den Graph, der entsteht, indem man die Mitte eines regulären 7-Ecks mit den Ecken verbindet. (Dieser Graph hat 8 Ecken und 14 Kanten.) Verwenden Sie CoCoA, um zu zeigen, dass dieser Graph keine 3-Färbung besitzt.

Aufgabe 20: (Das 8-Damenproblem) Sei ein n × n-Schachbrett wie in der Vorlesung durch Unbestimmte (x_{ij})_{i,j=1,...,n} beschrieben.

a) Schreiben Sie eine CoCoA-Funktion IsMove(…), die zwei Felder (d.h. zwei Unbestimmte) nimmt und prüft, ob eine Dame von dem einen Feld auf das andere ziehen kann.

b) Implementieren Sie eine CoCoA-Funktion QueenIdeal(…), die das Ideal berechnet, das allen möglichen Damenzügen auf einem n × n-Schachbrett entspricht.

c) Schreiben Sie eine CoCoA-Funktion Puzzle(…), die die maximale Anzahl N von Damen berechnet, die man auf ein n × n-Schachbrett stellen kann, ohne dass sich zwei gegenseitig attackieren, und die die Anzahl aller möglichen Lösungen für N Damen ausgibt.

d) Prüfen Sie Ihre Funktionen, indem Sie direkt zeigen, dass sich für n = 4 die Zahl

N = 4 ergibt und dass nur die folgenden beiden Möglichkeiten existieren:

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Lösen Sie die Aufgabe auch für möglichst viele n = 3, 4, 5, …, insbesondere für den klassischen Fall n = 8.