Gröbner Bases Tutorial - Computer Science - Amherst College

61 Pages·2007·0.62 MB·English

by  David A. Cox

Gröbner
Bases Tutorial
David A. Cox
Graph Theory
Gröbner Bases Tutorial
Geometric
Theorem
Part II: A Sampler of Recent Developments
Discovery
The Generic
GröbnerWalk
Phylogenetic
David A. Cox
Invariants
Department of Mathematics and Computer Science
Amherst College
(cid:0) (cid:1) (cid:2) (cid:3) (cid:2) (cid:4) (cid:5) (cid:1) (cid:6) (cid:7) (cid:8) (cid:9) (cid:4) (cid:10) (cid:5) (cid:8) (cid:0) (cid:11)
ISSAC 2007 Tutorial
Outline
Gröbner
Bases Tutorial
The original plan was to cover five topics:
David A. Cox
Graph Theory • Graph Theory
Geometric
• Geometric Theorem Discovery
Theorem
Discovery
• The Generic Gröbner Walk
The Generic
• Alternatives to the Buchberger Algorithm ← Too hard
GröbnerWalk
• Moduli of Quiver Representations ← Too complicated
Phylogenetic
Invariants
The new plan is to cover four topics:
• Graph Theory
• Geometric Theorem Discovery
• The Generic Gröbner Walk
• (New) Phylogenetic Invariants
Graph Colorings
Gröbner Let G = (V ,E) be a graph with vertices V = {1,...,n}.
Bases Tutorial
David A. Cox Definition
A k-coloring of G is a function from V to a set of k colors
Graph Theory
Geometric such that adjacent vertices have distinct colors.
Theorem
Discovery
Example
The Generic
GröbnerWalk
vertices = 81 squares
3 5
Phylogenetic
Invariants
edges = links between: 1 2 9
• squares in same column 7 6 2
• squares in same row 6 5 3
2 4 9
• squares in same 3 × 3
3 1 5
Colors = {1,2,...,9}
3 4 8
4 6 7
Goal: Extend the partial
3 1
coloring to a full coloring.
Graph Ideal
Gröbner
Definition
Bases Tutorial
David A. Cox C
The k-coloring ideal of G is the ideal I ⊆ [x | i ∈ V ]
G,k i
generated by:
Graph Theory
Geometric
Theorem k
for all i ∈ V : x − 1
Discovery
i
The Generic k−1 k−2 k−2 k−1
for all ij ∈ E : x + x x + ···+ x x + x .
GröbnerWalk i i j i j j
Phylogenetic
Invariants
Lemma
Cn
V(I ) ⊆ consists of all k-colorings of G for the set of
G,k
th
colors consisting of the k roots of unity
p
m z z 2 z k−1 z 2 i/k
= {1, , , ... , }, = e .
n k k k k
k k
(x − 1) − (x − 1)
i j
k−1 k−2 k−1
Proof. = x + x x + ··· + x .
j
x − x i i j
i j
The Existence of Colorings
Gröbner
Two Observations
Bases Tutorial
David A. Cox
G has a k-coloring ⇐⇒ V(I ) 6= 0/.
G,k
Graph Theory Hence the Consistency Theorem gives a Gröbner basis
Geometric
criterion for the existence of a k-coloring.
Theorem
Discovery
The Generic
3-Colorings
GröbnerWalk
Phylogenetic For 3-colorings, the ideal I is generated by
Invariants G,3
3
for all i ∈ V : x − 1
i
2 2
for all ij ∈ E : x + x x + x .
i i j j
These equations can be hard to solve!
Theorem
3-colorability is NP-complete.
Example
Gröbner This example of a graph
1 2
Bases Tutorial
with a 3-coloring is due
David A. Cox
to Chao and Chen (1993).
5 6
Graph Theory
Hillar and Windfeldt (2006) 12 7
Geometric
Theorem
compute the reduced Gröbner
Discovery 11 8
basis of the graph ideal I
The Generic
G,3 10 9
GröbnerWalk
for lex with x > ··· > x .
1 12
Phylogenetic
Invariants
4 3
The reduced Gröbner basis is:
3
{x − 1, x − x , x − x , x − x ,
12 7 12 4 12 3 12
2 2
x + x x + x , x − x , x − x , x − x ,
11 11 12 12 9 11 6 11 2 11
x + x + x , x + x + x , x + x + x ,
10 11 12 8 11 12 5 11 12
x + x + x }.
1 11 12
Note x − x , x − x , x − x ∈ I .
8 10 5 10 1 10 G,3
Uniquely k -Colorable Graphs
Gröbner
The Chao/Chen graph has essentially only one 3-coloring.
Bases Tutorial
David A. Cox
Definition
Graph Theory
A graph G is uniquely k-colorable if it has a unique
Geometric
Theorem k-coloring up the permutation of the colors.
Discovery
The Generic
Hillar and Windfeldt show that unique k-colorability is easy
GröbnerWalk
to detect using Gröbner bases.
Phylogenetic
Invariants
We start with a k-coloring of G that uses all k colors.
Assume the k colors occur among the last k vertices. Then:
Use variables x ,...,x , y ,...,y with lex order
1 n−k 1 k
x > ··· > x > y > ··· > y .
1 n−k 1 k
Use these variables to label the vertices of G.
Some Interesting Polynomials
Gröbner
Consider the following polynomials:
Bases Tutorial
David A. Cox
k
y − 1
k
Graph Theory
a a
h (y ,...,y ) = (cid:229) y j ···y k , j = 1,...,k − 1
Geometric j j k a +···+a =j j k
Theorem j k
Discovery
x − y , color(x ) = color(y ), j ≥ 2
i j i j
The Generic
GröbnerWalk x + y + ··· + y , color(x ) = color(y ).
i 2 k i 1
Phylogenetic
Invariants
In this notation, the Gröbner basis given earlier is:
3
{y − 1,
3
2 2
h (y ,y ) = y + y y + y , h (y ,y ,y ) = y + y + y ,
2 2 3 2 2 3 3 1 1 2 3 1 2 3
x − y , x − y , x − y , x − y , x − y , x − y ,
7 3 4 3 3 3 9 2 6 2 2 2
x + y + y , x + y + y , x + y + y }.
8 2 3 5 2 3 1 2 3
A Theorem
Gröbner Summary:
Bases Tutorial
G has vertices x ,...,x , y ,...,y .
David A. Cox 1 n−k 1 k
G has a k-coloring where y ,...,y get all the colors.
1 k
Graph Theory
C
[x,y] has lex with x > ··· > x > y > ··· > y .
Geometric 1 n−k 1 k
Theorem
Using this data, we create:
Discovery
The Generic C
The coloring ideal I ⊆ [x,y].
GröbnerWalk G,k
The n polynomials g ,...,g given by
Phylogenetic
1 n
Invariants
k
y − 1, h (y ,...,y ), x − y , x + y + ··· + y ,
j j k i j i 2 k
k
Theorem
The following are equivalent:
G is uniquely k-colorable.
g ,...,g ∈ I .
1 n G,k
{g ,...,g } is the reduced Gröbner basis for I .
1 n G,k
Remarks
Gröbner
Bases Tutorial
When G is uniquely k-colorable, the theorem implies
David A. Cox
Graph Theory k k−1 2
h (I )i = hy , y , ... , y , y , x , ... , x i.
LT
G,k k k−1 2 1 n−k 1
Geometric
Theorem
Discovery Since
The Generic
C
dim [x,y]/I = #monomials not in h (I )i, and
GröbnerWalk LT
G,k G,k
Phylogenetic I is radical,
Invariants G,k
a uniquely k-colorable graph has
C
#k-colorings = dim [x,y]/I = k · (k − 1)···2 · 1 = k!.
G,k
Hillar and Windfeldt have a version of this result that doesn’t
assume we know a k-coloring in advance.