Gröbner Bases Tutorial - Computer Science - Amherst College PDF
Preview Gröbner Bases Tutorial - Computer Science - Amherst College
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.
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