Lecture 1: Gröbner Bases
and Border Bases
— The Schizophrenic Lecture —
Martin Kreuzer
Fakultät für Informatik und Mathematik
Universität Passau
martin.kreuzer @ uni-passau.de
Sophus Lie Center
Nordfjordeid
June 15, 2009
1
Contents
2
Contents
1. Gröbner Bases
2-a
Contents
1. Gröbner Bases
2. Border Bases
2-b
Contents
1. Gröbner Bases
2. Border Bases
3. Properties of GB and BB
2-c
Contents
1. Gröbner Bases
2. Border Bases
3. Properties of GB and BB
4. Division Algorithms
2-d
Contents
1. Gröbner Bases
2. Border Bases
3. Properties of GB and BB
4. Division Algorithms
5. Neighbors
2-e
Contents
1. Gröbner Bases
2. Border Bases
3. Properties of GB and BB
4. Division Algorithms
5. Neighbors
6. The Buchberger Criterion
2-f
1 – Gröbner Bases
Before you criticize someone
you should walk a mile in their shoes.
3
1 – Gröbner Bases
Before you criticize someone
you should walk a mile in their shoes.
In this way, when you criticize them,
3-a
1 – Gröbner Bases
Before you criticize someone
you should walk a mile in their shoes.
In this way, when you criticize them,
you are a mile away
3-b
1 – Gröbner Bases
Before you criticize someone
you should walk a mile in their shoes.
In this way, when you criticize them,
you are a mile away
and you have their shoes.
3-c
1 – Gröbner Bases
Before you criticize someone
you should walk a mile in their shoes.
In this way, when you criticize them,
you are a mile away
and you have their shoes.
K field
P = K[x1 , . . . , xn ] polynomial ring over K
3-d
1 – Gröbner Bases
Before you criticize someone
you should walk a mile in their shoes.
In this way, when you criticize them,
you are a mile away
and you have their shoes.
K field
P = K[x1 , . . . , xn ] polynomial ring over K
αn
1
Tn = {xα
1 · · · xn | αi ≥ 0} monoid of terms
σ term ordering on Tn (complete, multiplicative well-ordering)
3-e
Definition of Gröbner Bases
(a) Every f ∈ P \ {0} has a unique representation
f = c1 t1 + · · · + cs ts with ci ∈ K \ {0} and ti ∈ Tn such that
t1 >σ · · · >σ ts . The term LTσ (f ) = t1 is called the leading term
of f and LCσ (f ) = c1 is the leading coefficient of f .
4
Definition of Gröbner Bases
(a) Every f ∈ P \ {0} has a unique representation
f = c1 t1 + · · · + cs ts with ci ∈ K \ {0} and ti ∈ Tn such that
t1 >σ · · · >σ ts . The term LTσ (f ) = t1 is called the leading term
of f and LCσ (f ) = c1 is the leading coefficient of f .
(b) For an ideal I ⊆ P , we let LTσ (I) = hLTσ (f ) | f ∈ I \ {0}i and
call it the leading term ideal of I.
4-a
Definition of Gröbner Bases
(a) Every f ∈ P \ {0} has a unique representation
f = c1 t1 + · · · + cs ts with ci ∈ K \ {0} and ti ∈ Tn such that
t1 >σ · · · >σ ts . The term LTσ (f ) = t1 is called the leading term
of f and LCσ (f ) = c1 is the leading coefficient of f .
(b) For an ideal I ⊆ P , we let LTσ (I) = hLTσ (f ) | f ∈ I \ {0}i and
call it the leading term ideal of I.
(c) A set of polynomials f1 , . . . , fs ∈ I is called a σ-Gröbner basis
of I if LTσ (I) = hLTσ (f1 ), . . . , LTσ (fs )i.
4-b
2 – Border Bases
Given the choice between two theories,
5
2 – Border Bases
Given the choice between two theories,
take the one which is funnier.
5-a
2 – Border Bases
Given the choice between two theories,
take the one which is funnier.
I ⊆ P zero-dimensional polynomial ideal (i.e. dimK (P/I) < ∞)
5-b
2 – Border Bases
Given the choice between two theories,
take the one which is funnier.
I ⊆ P zero-dimensional polynomial ideal (i.e. dimK (P/I) < ∞)
Open Problem: Give a good definition of border bases for
higher-dimensional ideals and generalize all results in this and the
subsequent lectures!
5-c
2 – Border Bases
Given the choice between two theories,
take the one which is funnier.
I ⊆ P zero-dimensional polynomial ideal (i.e. dimK (P/I) < ∞)
Open Problem: Give a good definition of border bases for
higher-dimensional ideals and generalize all results in this and the
subsequent lectures!
Definition 2.1 (a) A (finite) set O ⊂ Tn is called an order ideal if
every term dividing a term in O is contained in O.
5-d
2 – Border Bases
Given the choice between two theories,
take the one which is funnier.
I ⊆ P zero-dimensional polynomial ideal (i.e. dimK (P/I) < ∞)
Open Problem: Give a good definition of border bases for
higher-dimensional ideals and generalize all results in this and the
subsequent lectures!
Definition 2.1 (a) A (finite) set O ⊂ Tn is called an order ideal if
every term dividing a term in O is contained in O.
(b) Let O be an order ideal. The set ∂O = (x1 O ∪ · · · ∪ xn O) \ O is
called the border of O.
5-e
Picture of an Order Ideal and its Border
6
Picture of an Order Ideal and its Border
• term in the order ideal
6-a
◦ term in the border
Definition 2.2 (a) Let O = {t1 , . . . , tµ } be an order ideal and
∂O = {b1 , . . . , bν } its border. A set of polynomials {g1 , . . . , gν } ⊂ I of
the form
µ
P
gj = bj −
cij ti
i=1
with cij ∈ K and ti ∈ O is called an O-border prebasis of I.
7
Definition 2.2 (a) Let O = {t1 , . . . , tµ } be an order ideal and
∂O = {b1 , . . . , bν } its border. A set of polynomials {g1 , . . . , gν } ⊂ I of
the form
µ
P
gj = bj −
cij ti
i=1
with cij ∈ K and ti ∈ O is called an O-border prebasis of I.
(b) An O-border prebasis of I is called an O-border basis of I if the
residue classes of the terms in O are a K-vector space basis of P/I.
7-a
Definition 2.2 (a) Let O = {t1 , . . . , tµ } be an order ideal and
∂O = {b1 , . . . , bν } its border. A set of polynomials {g1 , . . . , gν } ⊂ I of
the form
µ
P
gj = bj −
cij ti
i=1
with cij ∈ K and ti ∈ O is called an O-border prebasis of I.
(b) An O-border prebasis of I is called an O-border basis of I if the
residue classes of the terms in O are a K-vector space basis of P/I.
Below se shall see that, given an O-border prebasis G, the set O is
always a system of generators of the K-vector space P/hGi.
7-b
Example of a Border Basis
Example 2.3 Consider the oder ideal O = {1, x, y, xy} in T2 . Its
border is ∂O = {x2 , x2 y, xy 2 , y 2 }.
8
Example of a Border Basis
Example 2.3 Consider the oder ideal O = {1, x, y, xy} in T2 . Its
border is ∂O = {x2 , x2 y, xy 2 , y 2 }.
y
....
......
◦
◦
•
•
◦
•
•
◦
1
8-a
................
x
Example of a Border Basis
Example 2.3 Consider the oder ideal O = {1, x, y, xy} in T2 . Its
border is ∂O = {x2 , x2 y, xy 2 , y 2 }.
y
....
......
◦
◦
•
•
◦
•
•
◦
1
................
x
The set of polynomials G = {g1 , g2 , g3 , g4 } where
g1 = x2 − x,
g2 = x2 y − xy,
is an O-border basis of I.
8-b
g3 = xy 2 − xy,
g4 = y 2 − y
3 – Properties of GB and BB
Martin’s Limerick:
9
3 – Properties of GB and BB
Martin’s Limerick:
The list of the theorems I knew
made limericks end at line two.
9-a
3 – Properties of GB and BB
Martin’s Limerick:
The list of the theorems I knew
made limericks end at line two.
Proposition 3.1 (Existence and Uniqueness of GB)
(a) For every term ordering σ and every ideal I ⊆ P , there exists a
σ-Gröbner basis of I.
9-b
3 – Properties of GB and BB
Martin’s Limerick:
The list of the theorems I knew
made limericks end at line two.
Proposition 3.1 (Existence and Uniqueness of GB)
(a) For every term ordering σ and every ideal I ⊆ P , there exists a
σ-Gröbner basis of I.
(b) A σ-Gröbner basis of I is a system of generators of I.
9-c
3 – Properties of GB and BB
Martin’s Limerick:
The list of the theorems I knew
made limericks end at line two.
Proposition 3.1 (Existence and Uniqueness of GB)
(a) For every term ordering σ and every ideal I ⊆ P , there exists a
σ-Gröbner basis of I.
(b) A σ-Gröbner basis of I is a system of generators of I.
(c) For every term ordering σ, an ideal I ⊆ P has a unique reduced
σ-Gröbner basis, i.e. a GB which is minimal, monic, and
completely interreduced.
9-d
For border bases, we shall always use the following notation.
10
For border bases, we shall always use the following notation.
O = {t1 , . . . , tµ } order ideal with border ∂O = {b1 , . . . , bν }
G = {g1 , . . . , gν } is an O-border prebasis, where
µ
P
gj = bj −
cij ti with cij ∈ K
i=1
10-a
For border bases, we shall always use the following notation.
O = {t1 , . . . , tµ } order ideal with border ∂O = {b1 , . . . , bν }
G = {g1 , . . . , gν } is an O-border prebasis, where
µ
P
gj = bj −
cij ti with cij ∈ K
i=1
Proposition 3.2 (Existence and Uniqueness of BB)
(a) Given an order ideal O, a 0-dimensional polynomial ideal I need
not have an O-border basis, even if #O = dimK (P/I).
10-b
For border bases, we shall always use the following notation.
O = {t1 , . . . , tµ } order ideal with border ∂O = {b1 , . . . , bν }
G = {g1 , . . . , gν } is an O-border prebasis, where
µ
P
gj = bj −
cij ti with cij ∈ K
i=1
Proposition 3.2 (Existence and Uniqueness of BB)
(a) Given an order ideal O, a 0-dimensional polynomial ideal I need
not have an O-border basis, even if #O = dimK (P/I).
(b) If a 0-dimensional ideal I ⊂ P has an O-border basis G then G
generates I.
10-c
For border bases, we shall always use the following notation.
O = {t1 , . . . , tµ } order ideal with border ∂O = {b1 , . . . , bν }
G = {g1 , . . . , gν } is an O-border prebasis, where
µ
P
gj = bj −
cij ti with cij ∈ K
i=1
Proposition 3.2 (Existence and Uniqueness of BB)
(a) Given an order ideal O, a 0-dimensional polynomial ideal I need
not have an O-border basis, even if #O = dimK (P/I).
(b) If a 0-dimensional ideal I ⊂ P has an O-border basis G then G
generates I.
(c) If a 0-dimensional ideal I ⊂ P has an O-border basis G then G is
uniquely determined.
10-d
The Relation Between GB and BB
Let σ be a term ordering. Then Oσ (I) = Tn \ LTσ (I) is an order
ideal of terms. By Macaulay’s Basis Theorem, the residue classes
of Oσ (I) form a K-basis of P/I.
11
The Relation Between GB and BB
Let σ be a term ordering. Then Oσ (I) = Tn \ LTσ (I) is an order
ideal of terms. By Macaulay’s Basis Theorem, the residue classes
of Oσ (I) form a K-basis of P/I.
Proposition 3.3 (Border Bases Generalize Gröbner Bases)
If O is of the form Tn \ LTσ (I) for some term ordering σ, then I has
an O-border basis.
It contains the reduced σ-Gröbner basis of I. The elements of the
reduced σ-GB are exactly the border basis polynomials corresponding
to the corners of ∂O, i.e. to the minimal generators of the border
term ideal.
11-a
4 – Division Algorithms
What is a proof ?
12
4 – Division Algorithms
What is a proof ?
One half percent of alcohol.
12-a
4 – Division Algorithms
What is a proof ?
One half percent of alcohol.
Theorem 4.1 (The Division Algorithm)
Let σ be a term ordering, f ∈ P , and G = (g1 , . . . , gν ) ∈ P ν .
Consider the following instructions:
D1. Let q1 = · · · = qν = 0, p = 0, and v = f .
D2. Find the smallest i ∈ {1, . . . , ν} such that LTσ (v) is a multiple of
LMσ (v)
and v by
LTσ (gi ). If such an i exists, replace qi by qi + LM
σ (gi )
v−
LMσ (v)
LMσ (gi )
· gi .
12-b
D3. Repeat step D2 until there is no more i ∈ {1, . . . , ν} such that
LTσ (v) is a multiple of LTσ (gi ). Then replace p by p + LMσ (v)
and v by v − LMσ (v).
D4. If now v 6= 0, start again with step D2. If v = 0, return the tuple
(q1 , . . . , qν ) ∈ P ν and p ∈ P .
13
D3. Repeat step D2 until there is no more i ∈ {1, . . . , ν} such that
LTσ (v) is a multiple of LTσ (gi ). Then replace p by p + LMσ (v)
and v by v − LMσ (v).
D4. If now v 6= 0, start again with step D2. If v = 0, return the tuple
(q1 , . . . , qν ) ∈ P ν and p ∈ P .
This is an algorithm which returns a tuple (q1 , . . . , qν ) ∈ P ν and
p ∈ P such that
f = q1 g1 + · · · + qν gν + p
such that Supp(p) ∩ hLTσ (g1 ), . . . , LTσ (gν )i = ∅, and such that
LTσ (qi gi ) ≤σ LTσ (m) if qi 6= 0.
13-a
D3. Repeat step D2 until there is no more i ∈ {1, . . . , ν} such that
LTσ (v) is a multiple of LTσ (gi ). Then replace p by p + LMσ (v)
and v by v − LMσ (v).
D4. If now v 6= 0, start again with step D2. If v = 0, return the tuple
(q1 , . . . , qν ) ∈ P ν and p ∈ P .
This is an algorithm which returns a tuple (q1 , . . . , qν ) ∈ P ν and
p ∈ P such that
f = q1 g1 + · · · + qν gν + p
such that Supp(p) ∩ hLTσ (g1 ), . . . , LTσ (gν )i = ∅, and such that
LTσ (qi gi ) ≤σ LTσ (m) if qi 6= 0.
Definition 4.2 The element NRσ,G (f ) = p is called the normal
remainder of f with respect to division by G.
13-b
Definition 4.3 Let O0 = O and Oi = Oi−1 ∪ ∂Oi−1 for i ≥ 1.
For every term t ∈ Tn , there is then a unique number i = indO (t) ≥ 0
such that t ∈ Oi \ Oi−1 . It is called the O-index of t.
Theorem 4.4 (The Border Division Algorithm)
Given a polynomial f , consider the following steps:
B1. Let f1 = · · · = fν = 0, c1 = · · · = cµ = 0, and h = f .
B2. If h = 0, then return (f1 , . . . , fν , c1 , . . . , cµ ) and stop.
B3. If indO (h) = 0, then find c1 , . . . , cµ ∈ K such that
h = c1 t1 + · · · + cµ tµ . Return (f1 , . . . , fν , c1 , . . . , cµ ) and stop.
14
B4. If indO (h) > 0, then let h = a1 h1 + · · · + as hs with
a1 , . . . , as ∈ K \ {0} and h1 , . . . , hs ∈ Tn such that
indO (h1 ) = indO (h). Determine the smallest index i ∈ {1, . . . , ν}
such that h1 factors as h1 = t0 bi with a term t0 of degree
indO (h) − 1. Subtract a1 t0 gi from h, add a1 t0 to fi , and continue
with step B2.
15
B4. If indO (h) > 0, then let h = a1 h1 + · · · + as hs with
a1 , . . . , as ∈ K \ {0} and h1 , . . . , hs ∈ Tn such that
indO (h1 ) = indO (h). Determine the smallest index i ∈ {1, . . . , ν}
such that h1 factors as h1 = t0 bi with a term t0 of degree
indO (h) − 1. Subtract a1 t0 gi from h, add a1 t0 to fi , and continue
with step B2.
This is an algorithm that returns a tuple
(f1 , . . . , fν , c1 , . . . , cµ ) ∈ P ν × K µ such that
f = f1 g1 + · · · + fν gν + c1 t1 + · · · + cµ tµ
and deg(fi ) ≤ indO (f ) − 1 for all i ∈ {1, . . . , ν} with fi gi 6= 0.
15-a
B4. If indO (h) > 0, then let h = a1 h1 + · · · + as hs with
a1 , . . . , as ∈ K \ {0} and h1 , . . . , hs ∈ Tn such that
indO (h1 ) = indO (h). Determine the smallest index i ∈ {1, . . . , ν}
such that h1 factors as h1 = t0 bi with a term t0 of degree
indO (h) − 1. Subtract a1 t0 gi from h, add a1 t0 to fi , and continue
with step B2.
This is an algorithm that returns a tuple
(f1 , . . . , fν , c1 , . . . , cµ ) ∈ P ν × K µ such that
f = f1 g1 + · · · + fν gν + c1 t1 + · · · + cµ tµ
and deg(fi ) ≤ indO (f ) − 1 for all i ∈ {1, . . . , ν} with fi gi 6= 0.
Corollary 4.5 The residue classes of the elements of O generate the
K-vector space P/I.
15-b
Gröbner Bases and Rewriting Systems
Let σ be a term ordering on Tn and G = {g1 , . . . , gν } ⊂ (P \ {0})ν .
16
Gröbner Bases and Rewriting Systems
Let σ be a term ordering on Tn and G = {g1 , . . . , gν } ⊂ (P \ {0})ν .
Definition 4.6 (a) Let f1 , f2 ∈ P , and suppose there exist a
constant c ∈ K, a term t ∈ Tn , and an index i ∈ {1, . . . , ν} such
that f2 = f1 − c tgi and t · LTσ (gi ) ∈
/ Supp(f2 ). Then we say that
gi
f1 reduces to f2 in one step, and we write f1 −→ f2 .
16-a
Gröbner Bases and Rewriting Systems
Let σ be a term ordering on Tn and G = {g1 , . . . , gν } ⊂ (P \ {0})ν .
Definition 4.6 (a) Let f1 , f2 ∈ P , and suppose there exist a
constant c ∈ K, a term t ∈ Tn , and an index i ∈ {1, . . . , ν} such
that f2 = f1 − c tgi and t · LTσ (gi ) ∈
/ Supp(f2 ). Then we say that
gi
f1 reduces to f2 in one step, and we write f1 −→ f2 .
g1
gs
(b) The transitive closure of the relations −→, . . . , −→ is called the
G
rewrite relation defined by G and is denoted by −→.
16-b
Gröbner Bases and Rewriting Systems
Let σ be a term ordering on Tn and G = {g1 , . . . , gν } ⊂ (P \ {0})ν .
Definition 4.6 (a) Let f1 , f2 ∈ P , and suppose there exist a
constant c ∈ K, a term t ∈ Tn , and an index i ∈ {1, . . . , ν} such
that f2 = f1 − c tgi and t · LTσ (gi ) ∈
/ Supp(f2 ). Then we say that
gi
f1 reduces to f2 in one step, and we write f1 −→ f2 .
g1
gs
(b) The transitive closure of the relations −→, . . . , −→ is called the
G
rewrite relation defined by G and is denoted by −→.
Proposition 4.7 A set of polynomials G = {g1 , . . . , gν } is a
G
σ-Gröbner basis if and only if the rewrite rule −→ is confluent.
G
G
This means that if there are reductions f1 −→ f2 and f1 −→ f3 then
G
G
there exist a polynomial f4 and reductions f2 −→ f4 and f3 −→ f4 .
16-c
Border Bases and Rewriting Systems
Proposition 4.8 Let G = {g1 , . . . , gν } be an O-border prebasis.
Then G is an O-border basis if and only if the rewriting system
µ
P
defined by the rules bj −→
cij ti is confluent.
i=1
17
Border Bases and Rewriting Systems
Proposition 4.8 Let G = {g1 , . . . , gν } be an O-border prebasis.
Then G is an O-border basis if and only if the rewriting system
µ
P
defined by the rules bj −→
cij ti is confluent.
i=1
Notice that this rewriting system is in general not terminating, i.e.
not Noetherian. This means that there may be an infinite sequences
of reductions
G
G
G
f1 −→ f2 −→ f3 −→ · · ·
17-a
5 – Neighbors
Give a man a fish and he will eat for a day.
Teach him how to fish, and he will
18
5 – Neighbors
Give a man a fish and he will eat for a day.
Teach him how to fish, and he will
sit in a boat and drink beer all day.
18-a
5 – Neighbors
Give a man a fish and he will eat for a day.
Teach him how to fish, and he will
sit in a boat and drink beer all day.
Definition 5.1 Let bi , bj ∈ ∂O be two distinct border terms.
18-b
5 – Neighbors
Give a man a fish and he will eat for a day.
Teach him how to fish, and he will
sit in a boat and drink beer all day.
Definition 5.1 Let bi , bj ∈ ∂O be two distinct border terms.
(a) The border terms bi and bj are called next-door neighbors if
bi = xk bj for some k ∈ {1, . . . , n}.
18-c
5 – Neighbors
Give a man a fish and he will eat for a day.
Teach him how to fish, and he will
sit in a boat and drink beer all day.
Definition 5.1 Let bi , bj ∈ ∂O be two distinct border terms.
(a) The border terms bi and bj are called next-door neighbors if
bi = xk bj for some k ∈ {1, . . . , n}.
(b) The border terms bi and bj are called across-the-street
neighbors if xk bi = x` bj for some k, ` ∈ {1, . . . , n}.
18-d
5 – Neighbors
Give a man a fish and he will eat for a day.
Teach him how to fish, and he will
sit in a boat and drink beer all day.
Definition 5.1 Let bi , bj ∈ ∂O be two distinct border terms.
(a) The border terms bi and bj are called next-door neighbors if
bi = xk bj for some k ∈ {1, . . . , n}.
(b) The border terms bi and bj are called across-the-street
neighbors if xk bi = x` bj for some k, ` ∈ {1, . . . , n}.
(c) The border terms bi and bj are called neighbors if they are
next-door neighbors or across-the-street neighbors.
18-e
Example 5.2 The border of O = {1, x, y, xy} is
∂O = {x2 , x2 y, xy 2 , y 2 }. Here the neighbor relations look as follows:
(x2 , x2 y) and (y 2 , xy 2 ) are next-door neighbor pairs
(x2 y, xy 2 ) is an across-the-street neighbor pair
y
.
........
◦...........................◦..............
•
•
•
•
1
.....
.....
.....
.....
.....
...
..
..
..
..
..
...
◦
◦
19
...............
x
Neighbor Syzygies
Definition 5.3 (a) For t, t0 ∈ Tn , we call the pair
(lcm(t, t0 )/t, − lcm(t, t0 )/t0 ) the fundamental syzygy of (t, t0 ).
20
Neighbor Syzygies
Definition 5.3 (a) For t, t0 ∈ Tn , we call the pair
(lcm(t, t0 )/t, − lcm(t, t0 )/t0 ) the fundamental syzygy of (t, t0 ).
(b) The fundamental syzygies of neighboring border terms are also
called the neighbor syzygies.
20-a
Neighbor Syzygies
Definition 5.3 (a) For t, t0 ∈ Tn , we call the pair
(lcm(t, t0 )/t, − lcm(t, t0 )/t0 ) the fundamental syzygy of (t, t0 ).
(b) The fundamental syzygies of neighboring border terms are also
called the neighbor syzygies.
Proposition 5.4 (a) Given a tuple of terms (t1 , . . . , tr ), the
fundamental syzygies σij = (lcm(ti , tj )/ti ) ei − (lcm(ti , tj )/tj ) ej such
that 1 ≤ i < j ≤ r generate the syzygy module
SyzP (t1 , . . . , tr ) = {(f1 , . . . , fr ) ∈ P r | f1 t1 + · · · + fr tr = 0}.
20-b
Neighbor Syzygies
Definition 5.3 (a) For t, t0 ∈ Tn , we call the pair
(lcm(t, t0 )/t, − lcm(t, t0 )/t0 ) the fundamental syzygy of (t, t0 ).
(b) The fundamental syzygies of neighboring border terms are also
called the neighbor syzygies.
Proposition 5.4 (a) Given a tuple of terms (t1 , . . . , tr ), the
fundamental syzygies σij = (lcm(ti , tj )/ti ) ei − (lcm(ti , tj )/tj ) ej such
that 1 ≤ i < j ≤ r generate the syzygy module
SyzP (t1 , . . . , tr ) = {(f1 , . . . , fr ) ∈ P r | f1 t1 + · · · + fr tr = 0}.
(b) The neighbor syzygies generate the module of border syzygies
SyzP (b1 , . . . , bν ).
20-c
6 – The Buchberger Criterion
What is higher mathematics?
21
6 – The Buchberger Criterion
What is higher mathematics?
If you awake in the morning with an unknown.
21-a
6 – The Buchberger Criterion
What is higher mathematics?
If you awake in the morning with an unknown.
Definition 6.1 Let gi , gj ∈ G be two distinct border prebasis
polynomials. Then the polynomial
Sij = (lcm(bi , bj )/bi ) · gi − (lcm(bi , bj )/bj ) · gj
is called the S-polynomial of gi and gj .
21-b
6 – The Buchberger Criterion
What is higher mathematics?
If you awake in the morning with an unknown.
Definition 6.1 Let gi , gj ∈ G be two distinct border prebasis
polynomials. Then the polynomial
Sij = (lcm(bi , bj )/bi ) · gi − (lcm(bi , bj )/bj ) · gj
is called the S-polynomial of gi and gj .
Theorem 6.2 (Stetter)
An O-border prebasis G is an O-border basis if and only if the
neighbor syzygies lift, i.e. if and only if we have NRO,G (Sij ) = 0 for
all (i, j) such that (bi , bj ) is a pair of neighbors.
21-c
Advantages of Border Bases
22
Advantages of Border Bases
1. Border bases are numerically stable.
If one changes the coefficicients of some polynomials generating I
slightly, the border basis of I changes continuously.
22-a
Advantages of Border Bases
1. Border bases are numerically stable.
If one changes the coefficicients of some polynomials generating I
slightly, the border basis of I changes continuously.
2. Border bases preserve symmetries.
There are many more order ideals O for which a given ideal I has
a border basis than order ideals of the form
Oσ (I) = Tn \ LTσ (I). Frequently, there are border bases having
the same symmetries as the initial generating system.
22-b
Advantages of Border Bases
1. Border bases are numerically stable.
If one changes the coefficicients of some polynomials generating I
slightly, the border basis of I changes continuously.
2. Border bases preserve symmetries.
There are many more order ideals O for which a given ideal I has
a border basis than order ideals of the form
Oσ (I) = Tn \ LTσ (I). Frequently, there are border bases having
the same symmetries as the initial generating system.
3. Border bases yield an explicit moduli space.
The cefficients of a border prebasis are parametrized by an affine
space. The border basis scheme is defined in this affine space
by explicit equations.
22-c