Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Geometry of Hilbert polynomials andGröbner bases
John Perry
Department of Mathematics, The University of Southern Mississippi
February 2010
Algebra is merely geometry in words;geometry is merely algebra in pictures.
— Sophie Germain
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Outline
1 Gröbner bases and monomial orderings
2 Hilbert polynomials
3 Caboara, Sturmfels’ algorithms
4 Future
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Outline
1 Gröbner bases and monomial orderings
2 Hilbert polynomials
3 Caboara, Sturmfels’ algorithms
4 Future
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Gröbner basis: definition from example
-4 -3 -2 -1 -0 1 2 3 4
-4
-3
-2
-1
-0
1
2
3
4
what can I say about • ?
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Monomial orderings
x2+ xy2
↙ ↘
x2+ xy x2+ xy2
(lex) (tdeg)
Remark
• notation: lm (p)• uncountably many orderings• given an ideal, finitely many equivalence classes
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Monomial orderings
x2+ xy2
↙ ↘
x2+ xy x2+ xy2
(lex) (tdeg)
Remark
• notation: lm (p)• uncountably many orderings• given an ideal, finitely many equivalence classes
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Gröbner basis: precise definition
G a Gröbner basis of ideal I?
〈lm(I)〉
1 2 3 4
1
2
3
4
• I = ⟨G⟩. . .
•
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Gröbner basis: precise definition
G a Gröbner basis of ideal I?
〈lm(G)〉
1 2 3 4
1
2
3
4
y3 6∈
lm (G)�
• I = ⟨G⟩ but•
lm (I)�
6=
lm (G)�
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Gröbner basis: precise definition
G a Gröbner basis of ideal I?
〈lm(G)〉
1 2 3 4
1
2
3
4
y3 ∈
lm (G)�
• I = ⟨G⟩ and•
lm (I)�
=
lm (G)�
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Gröbner basis: facts
• finite GB exists for any I• GB varies by equivalence class
• size, density, complexity• unique “reduced GB”
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Computing Gröbner bases
Buchberger’s algorithm (1965), others• while
lm (G)�
6=
lm (I)�
:
1 2 3 4
1
2
3
4
add • to G
• • “easy” iff
lm (G)�
6=
lm (I)�
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Minkowski sum of Newton polyhedra
another view of orderings
F =�
x2+ y2− 4,xy− 1
1 2 3
1
2
3
(vertex← exponent vector of each u ∈ f1 · · · fm)
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Big-time fact
Normal cones of Minkowski sum of Fl
equivalence classes
1 2 3 4
1
2
3
4
classification of orderings←→ discrete geometry
(Gritzmann and Sturmfels, 1993)
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Outline
1 Gröbner bases and monomial orderings
2 Hilbert polynomials
3 Caboara, Sturmfels’ algorithms
4 Future
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Hilbert function: definition from example
-2.5 -2 -1.5 -1 -0.5 -0 0.5 1 1.5 2 2.5
-2.5
-2
-1.5
-1
-0.5
-0
0.5
1
1.5
2
2.5
dimF (•)?
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Hilbert function: precise definition
R= F�
x1, . . . ,xn�
, homogeneous ideal
Hilbert function:
HFI :N→Nd→ dimF (Rd/Id) .
Facts
• ∃HFI for all I• HFI =HF⟨lm(I)⟩
1 2 3 4
1
2
3
4
dimF (Rd/Id) = dimF�
R/
lm (I)��
d
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Hilbert polynomial
Hilbert polynomial:
HPI (d) =HFI (d) for “sufficiently large” d.
1 2 3 4
1
2
3
4
Facts
• ∃HFI ,HPI for all I• degHPI = dimF (•)
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
HFI (d) ¡ GB (I)
Know G=GB (I), not HFI?• HF⟨lm(G)⟩ =HFI
• “easy” to compute:
• HSI , power series expansion of HFI• HPI
Know HFI , not GB (I)?• HFI (d) degree d mononomials not in
lm (G)�
d?
• (((((((degree d pairs
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
HFI (d) ¡ GB (I)
Know G=GB (I), not HFI?• HF⟨lm(G)⟩ =HFI
• “easy” to compute:
• HSI , power series expansion of HFI• HPI
Know HFI , not GB (I)?• HFI (d) degree d mononomials not in
lm (G)�
d?
• (((((((degree d pairs
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Outline
1 Gröbner bases and monomial orderings
2 Hilbert polynomials
3 Caboara, Sturmfels’ algorithms
4 Future
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Problem statement
• GB varies by equivalence class
• size, density, complexity• some orderings more efficient than others
Is it advisable to change orderings while computing a Gröbnerbasis?
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
“Tentative” Hilbert functions
J =¬
lm�
Gh�¶
Definitiontentative Hilbert function of ⟨G⟩ is HFJ
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Example 1
• Gh =�
x2+ y2− 4h2,xy− h2
• lm�
Gh�
=�
x2,xy
HSJ (t) =t2− t− 1
−t2+ 2t− 1HPJ (d) = d+ 2
HSI (d) =t2+ 2t+ 1
−t+ 1HPI (d) = 4
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Example 1
• Gh =�
x2+ y2− 4h2,xy− h2
• lm�
Gh�
=�
x2,xy
HSJ (t) =t2− t− 1
−t2+ 2t− 1HPJ (d) = d+ 2
HSI (d) =t2+ 2t+ 1
−t+ 1HPI (d) = 4
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Example 2
G=�
x3y2+ 5776x2y3+ · · ·,x4y+ · · ·
tdeg, x> y tdeg, x< y
HPJ (d) = 4d− 1 HPJ (d) = 3d+ 4
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Which ordering “better”?
If G not GB,
1 2 3 4 5
1
2
3
4
5
1 2 3 4 5
1
2
3
4
5
R/
lm (G)�
R/
lm (I)�
HFI (d) = dimF (R/I)d
∴ minimize HF⟨lm(G)⟩ (d) in long run
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Example
dHPI 0 1 2 3 4 5 6 #GB
3d+ 4 4 7 10 13 16 19 22 54d− 1 -1 3 7 11 15 19 23 15
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Gritzmann-Sturmfels algorithm
“Dynamic” Buchberger algorithm:• after adding new polynomial:
• compute HF for each normal cone• select ordering w/“best” HF
Gritzmann-Sturmfels, 1993
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Caboara algorithm
Gritzmann-Sturmfels algorithm:• compute HF for each normal subcone
Observations• trade-off in complexity
• most timings improve• some worsen
• only implementation of Gritzmann-Sturmfels algorithm?
Caboara, 1993
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Caboara algorithm
Gritzmann-Sturmfels algorithm:• compute HF for each normal subcone
Observations• trade-off in complexity
• most timings improve• some worsen
• only implementation of Gritzmann-Sturmfels algorithm?
Caboara, 1993
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Outline
1 Gröbner bases and monomial orderings
2 Hilbert polynomials
3 Caboara, Sturmfels’ algorithms
4 Future
Geometry ofHilbert
polynomialsand Gröbner
bases
John Perry
Gröbner basesand monomialorderingsGröbner bases
Geometry of orderings
Hilbertpolynomials
Caboara,Sturmfels’algorithms
Future
Open question
• small penalty to compute HF• diminishing benefit from “better” order• when quit computing HF?• guess: when HFJ (d) stable up to largest degree pair?