Geometry of Hilbert polynomials and Gröbner bases of Hilbert polynomials and Gröbner bases John Perry Gröbner bases and monomial orderings Gröbner bases Geometry of orderings Hilbert

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


bases

John Perry



Hilbertpolynomials


Future

Outline

1 Gröbner bases and monomial orderings

2 Hilbert polynomials

3 Caboara, Sturmfels’ algorithms

4 Future

Geometry ofHilbert


bases

John Perry



Hilbertpolynomials


Future

Outline




4 Future

Geometry ofHilbert


bases

John Perry



Hilbertpolynomials


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


bases

John Perry



Hilbertpolynomials


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


bases

John Perry



Hilbertpolynomials


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


bases

John Perry



Hilbertpolynomials


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


bases

John Perry



Hilbertpolynomials


Future



〈lm(G)〉

1 2 3 4

1

2

3

4

y3 6∈

lm (G)�

• I = ⟨G⟩ but•

lm (I)�

6=

lm (G)�

Geometry ofHilbert


bases

John Perry



Hilbertpolynomials


Future



〈lm(G)〉

1 2 3 4

1

2

3

4

y3 ∈

lm (G)�

• I = ⟨G⟩ and•

lm (I)�

=

lm (G)�

Geometry ofHilbert


bases

John Perry



Hilbertpolynomials


Future

Gröbner basis: facts

• finite GB exists for any I• GB varies by equivalence class

• size, density, complexity• unique “reduced GB”

Geometry ofHilbert


bases

John Perry



Hilbertpolynomials


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


bases

John Perry



Hilbertpolynomials


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


bases

John Perry



Hilbertpolynomials


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


bases

John Perry



Hilbertpolynomials


Future

Outline




4 Future

Geometry ofHilbert


bases

John Perry



Hilbertpolynomials


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


bases

John Perry



Hilbertpolynomials


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


bases

John Perry



Hilbertpolynomials


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


bases

John Perry



Hilbertpolynomials


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


bases

John Perry



Hilbertpolynomials


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


bases

John Perry



Hilbertpolynomials


Future

Outline




4 Future

Geometry ofHilbert


bases

John Perry



Hilbertpolynomials


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


bases

John Perry



Hilbertpolynomials


Future

“Tentative” Hilbert functions

J =¬

lm�

Gh�¶

Definitiontentative Hilbert function of ⟨G⟩ is HFJ

Geometry ofHilbert


bases

John Perry



Hilbertpolynomials


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


bases

John Perry



Hilbertpolynomials


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


bases

John Perry



Hilbertpolynomials


Future

Example 2

G=�

x3y2+ 5776x2y3+ · · ·,x4y+ · · ·

tdeg, x> y tdeg, x< y

HPJ (d) = 4d− 1 HPJ (d) = 3d+ 4

Geometry ofHilbert


bases

John Perry



Hilbertpolynomials


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


bases

John Perry



Hilbertpolynomials


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


bases

John Perry



Hilbertpolynomials


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


bases

John Perry



Hilbertpolynomials


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


bases

John Perry



Hilbertpolynomials


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


bases

John Perry



Hilbertpolynomials


Future

Outline




4 Future

Geometry ofHilbert


bases

John Perry



Hilbertpolynomials


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?

Geometry ofHilbert


bases

John Perry



Hilbertpolynomials


Future

Finis

Thank you!

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Geometry of Hilbert polynomials and Gröbner bases of Hilbert polynomials and Gröbner bases John Perry Gröbner bases and monomial orderings Gröbner bases Geometry of orderings Hilbert

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