[PPT] - For an (associative but not necessarily commutative) ring with PowerPoint Presentation

SLIDE 1

BUCHBERGER THEORY FOR EFFECTIVE ASSOCIATIVE RINGS

T. M., Seven variations on standard bases, (1988)

A solution if the ring is a vectorspace over a field APEL J., Computational ideal theory in finitely generated extension rings, T.C.S. 224 (2000), 1–33 Extension to suitable rings which are algebra over a ring Gateva, Weispfenning and Passau group, Reinert, . . .

SLIDE 2

BUCHBERGER THEORY FOR EFFECTIVE ASSOCIATIVE RINGS

T. M., Seven variations on standard bases, (1988)

A solution if the ring is a vectorspace over a field APEL J., Computational ideal theory in finitely generated extension rings, T.C.S. 224 (2000), 1–33 Extension to suitable rings which are algebra over a ring Gateva, Weispfenning and Passau group, Reinert, . . .

SLIDE 3

BUCHBERGER THEORY FOR EFFECTIVE ASSOCIATIVE RINGS

T. M., Seven variations on standard bases, (1988)

A solution if the ring is a vectorspace over a field APEL J., Computational ideal theory in finitely generated extension rings, T.C.S. 224 (2000), 1–33 Extension to suitable rings which are algebra over a ring Gateva, Weispfenning and Passau group, Reinert, . . .

SLIDE 4

BUCHBERGER THEORY FOR EFFECTIVE ASSOCIATIVE RINGS

T. M., Seven variations on standard bases, (1988)

A solution if the ring is a vectorspace over a field APEL J., Computational ideal theory in finitely generated extension rings, T.C.S. 224 (2000), 1–33 Extension to suitable rings which are algebra over a ring Gateva, Weispfenning and Passau group, Reinert, . . .

SLIDE 5

THEOREM For an (associative but not necessarily commutative) ring with identity A, there is a (not necessarily finite nor necessarily countable) set Z and a projection Π : Q := ZZ ։ A so that, denoting I ⊂ Q = ZZ the bilateral ideal I := ker(Π), we have A = Q/I.

SLIDE 6

EFFECTIVELY GIVEN ASSOCIATIVE RINGS

Let R be an (associative but not necessarily commutative) ring with identity 1R and A another (associative but not necessarily commutative) ring with identity 1A which is a left module on R. We consider A to be effectively given when we are given

sets v := {x1, . . . , xj, . . .}, V := {X1, . . . , Xi, . . .}, which are

countable and

Z := v ⊔ V = {x1, . . . , xj, . . . , X1, . . . , Xi, . . .};
rings R := Zv ⊂ Q := ZZ;
projections π : R = Zx1, . . . , xj, . . . ։ R and
Π : Q := Zx1, . . . , xj, . . . , X1, . . . , Xi, . . . ։ A which satisfies

Π(xj) = π(xj)1A, for each xj ∈ v, so that Π (R) = {r1A : r ∈ R} ⊂ A.

SLIDE 7

EFFECTIVELY GIVEN ASSOCIATIVE RINGS

Let R be an (associative but not necessarily commutative) ring with identity 1R and A another (associative but not necessarily commutative) ring with identity 1A which is a left module on R. We consider A to be effectively given when we are given

sets v := {x1, . . . , xj, . . .}, V := {X1, . . . , Xi, . . .}, which are

countable and

Z := v ⊔ V = {x1, . . . , xj, . . . , X1, . . . , Xi, . . .};
rings R := Zv ⊂ Q := ZZ;
projections π : R = Zx1, . . . , xj, . . . ։ R and
Π : Q := Zx1, . . . , xj, . . . , X1, . . . , Xi, . . . ։ A which satisfies

Π(xj) = π(xj)1A, for each xj ∈ v, so that Π (R) = {r1A : r ∈ R} ⊂ A.

SLIDE 8

EFFECTIVELY GIVEN ASSOCIATIVE RINGS

Let R be an (associative but not necessarily commutative) ring with identity 1R and A another (associative but not necessarily commutative) ring with identity 1A which is a left module on R. We consider A to be effectively given when we are given

sets v := {x1, . . . , xj, . . .}, V := {X1, . . . , Xi, . . .}, which are

countable and

Z := v ⊔ V = {x1, . . . , xj, . . . , X1, . . . , Xi, . . .};
rings R := Zv ⊂ Q := ZZ;
projections π : R = Zx1, . . . , xj, . . . ։ R and
Π : Q := Zx1, . . . , xj, . . . , X1, . . . , Xi, . . . ։ A which satisfies

Π(xj) = π(xj)1A, for each xj ∈ v, so that Π (R) = {r1A : r ∈ R} ⊂ A.

SLIDE 9

EFFECTIVELY GIVEN ASSOCIATIVE RINGS

Let R be an (associative but not necessarily commutative) ring with identity 1R and A another (associative but not necessarily commutative) ring with identity 1A which is a left module on R. We consider A to be effectively given when we are given

sets v := {x1, . . . , xj, . . .}, V := {X1, . . . , Xi, . . .}, which are

countable and

Z := v ⊔ V = {x1, . . . , xj, . . . , X1, . . . , Xi, . . .};
rings R := Zv ⊂ Q := ZZ;
projections π : R = Zx1, . . . , xj, . . . ։ R and
Π : Q := Zx1, . . . , xj, . . . , X1, . . . , Xi, . . . ։ A which satisfies

Π(xj) = π(xj)1A, for each xj ∈ v, so that Π (R) = {r1A : r ∈ R} ⊂ A.

SLIDE 10

EFFECTIVELY GIVEN ASSOCIATIVE RINGS

Π : Q := Zx1, . . . , xj, . . . , X1, . . . , Xi, . . . ։ A π : R = Zx1, . . . , xj, . . . ։ R Thus denoting

I := ker(Π) ⊂ Q and
I := I ∩ R = ker(π) ⊂ R,

we have A = Q/I and R = R/I; moreover we can wlog assume that R ⊂ A. Q as Z-module: using as alphabet V all symbols representing the primes. Further, when considering A as effectively given in this way, we explicitly require that Xixj ≡

i

l=1

π(alij)Xl + π(a0ij) mod I, alij ∈ Zv, ∀Xi ∈ V, xj ∈ v. If not ZX, Y as left Z[X]-module requires 1 ≥ X ≥ X 2

SLIDE 11

EFFECTIVELY GIVEN ASSOCIATIVE RINGS

Π : Q := Zx1, . . . , xj, . . . , X1, . . . , Xi, . . . ։ A π : R = Zx1, . . . , xj, . . . ։ R Thus denoting

I := ker(Π) ⊂ Q and
I := I ∩ R = ker(π) ⊂ R,

we have A = Q/I and R = R/I; moreover we can wlog assume that R ⊂ A. Q as Z-module: using as alphabet V all symbols representing the primes. Further, when considering A as effectively given in this way, we explicitly require that Xixj ≡

i

l=1

π(alij)Xl + π(a0ij) mod I, alij ∈ Zv, ∀Xi ∈ V, xj ∈ v. If not ZX, Y as left Z[X]-module requires 1 ≥ X ≥ X 2

SLIDE 12

EFFECTIVELY GIVEN ASSOCIATIVE RINGS

Π : Q := Zx1, . . . , xj, . . . , X1, . . . , Xi, . . . ։ A π : R = Zx1, . . . , xj, . . . ։ R Thus denoting

I := ker(Π) ⊂ Q and
I := I ∩ R = ker(π) ⊂ R,

we have A = Q/I and R = R/I; moreover we can wlog assume that R ⊂ A. Q as Z-module: using as alphabet V all symbols representing the primes. Further, when considering A as effectively given in this way, we explicitly require that Xixj ≡

i

l=1

π(alij)Xl + π(a0ij) mod I, alij ∈ Zv, ∀Xi ∈ V, xj ∈ v. If not ZX, Y as left Z[X]-module requires 1 ≥ X ≥ X 2

SLIDE 13

EFFECTIVELY GIVEN ASSOCIATIVE RINGS

Π : Q := Zx1, . . . , xj, . . . , X1, . . . , Xi, . . . ։ A π : R = Zx1, . . . , xj, . . . ։ R Thus denoting

I := ker(Π) ⊂ Q and
I := I ∩ R = ker(π) ⊂ R,

we have A = Q/I and R = R/I; moreover we can wlog assume that R ⊂ A. Q as Z-module: using as alphabet V all symbols representing the primes. Further, when considering A as effectively given in this way, we explicitly require that Xixj ≡

i

l=1

π(alij)Xl + π(a0ij) mod I, alij ∈ Zv, ∀Xi ∈ V, xj ∈ v. If not ZX, Y as left Z[X]-module requires 1 ≥ X ≥ X 2

SLIDE 14

EFFECTIVELY GIVEN ASSOCIATIVE RINGS

Π : Q := Zx1, . . . , xj, . . . , X1, . . . , Xi, . . . ։ A = Q/I π : R = Zx1, . . . , xj, . . . ։ R = R/I If we fix a term-ordering < on Z we can assume I to be given via its Gr¨

bner basis G w.r.t. < and, if < satisfies

Xi > t for each t ∈ v and Xi ∈ V also I is given via its Gr¨

bner basis G0 := G ∩ R w.r.t. <.

SLIDE 15

EFFECTIVELY GIVEN ASSOCIATIVE RINGS

Π : Q := Zx1, . . . , xj, . . . , X1, . . . , Xi, . . . ։ A = Q/I π : R = Zx1, . . . , xj, . . . ։ R = R/I If we fix a term-ordering < on Z we can assume I to be given via its Gr¨

bner basis G w.r.t. < and, if < satisfies

Xi > t for each t ∈ v and Xi ∈ V also I is given via its Gr¨

bner basis G0 := G ∩ R w.r.t. <.

SLIDE 16

EFFECTIVELY GIVEN ASSOCIATIVE RINGS

Π : Q := Zx1, . . . , xj, . . . , X1, . . . , Xi, . . . ։ A = Q/I π : R = Zx1, . . . , xj, . . . ։ R = R/I If we fix a term-ordering < on Z we can assume I to be given via its Gr¨

bner basis G w.r.t. < and, if < satisfies

Xi > t for each t ∈ v and Xi ∈ V also I is given via its Gr¨

bner basis G0 := G ∩ R w.r.t. <.

SLIDE 17

EFFECTIVELY GIVEN ASSOCIATIVE RINGS

Π : Q := Zx1, . . . , xj, . . . , X1, . . . , Xi, . . . ։ A = Q/I π : R = Zx1, . . . , xj, . . . ։ R = R/I If we fix a term-ordering < on Z we can assume I to be given via its Gr¨

bner basis G w.r.t. < and, if < satisfies

Xi > t for each t ∈ v and Xi ∈ V also I is given via its Gr¨

bner basis G0 := G ∩ R w.r.t. <.

Xixj ≡

i

l=1

π(alij)Xl + π(a0ij) mod I, alij ∈ Zv, ∀Xi ∈ V, xj ∈ v

SLIDE 18

EFFECTIVELY GIVEN ASSOCIATIVE RINGS

Π : Q := Zx1, . . . , xj, . . . , X1, . . . , Xi, . . . ։ A = Q/I π : R = Zx1, . . . , xj, . . . ։ R = R/I If we fix a term-ordering < on Z we can assume I to be given via its Gr¨

bner basis G w.r.t. < and, if < satisfies

Xi > t for each t ∈ v and Xi ∈ V also I is given via its Gr¨

bner basis G0 := G ∩ R w.r.t. <.

fij := Xixj −

i

l=1

π(alij)Xl − π(a0ij) ∈ I, alij ∈ Zv, ∀Xi ∈ V, xj ∈ v

SLIDE 19

EFFECTIVELY GIVEN ASSOCIATIVE RINGS

Π : Q := Zx1, . . . , xj, . . . , X1, . . . , Xi, . . . ։ A = Q/I π : R = Zx1, . . . , xj, . . . ։ R = R/I If we fix a term-ordering < on Z we can assume I to be given via its Gr¨

bner basis G w.r.t. < and, if < satisfies

Xi > t for each t ∈ v and Xi ∈ V also I is given via its Gr¨

bner basis G0 := G ∩ R w.r.t. <.

fij := Xixj −

i

l=1

π(alij)Xl − π(a0ij) ∈ I, alij ∈ Zv, ∀Xi ∈ V, xj ∈ v Xixj = T(fij) for each Xi ∈ V, xj ∈ v

SLIDE 20

EFFECTIVELY GIVEN ASSOCIATIVE RINGS

Π : Q := Zx1, . . . , xj, . . . , X1, . . . , Xi, . . . ։ A = Q/I π : R = Zx1, . . . , xj, . . . ։ R = R/I If we fix a term-ordering < on Z we can assume I to be given via its Gr¨

bner basis G w.r.t. < and, if < satisfies

Xi > t for each t ∈ v and Xi ∈ V also I is given via its Gr¨

bner basis G0 := G ∩ R w.r.t. <.

fij := Xixj −

i

l=1

π(alij)Xl − π(a0ij) ∈ I, alij ∈ Zv, ∀Xi ∈ V, xj ∈ v

Xixj for each Xi ∈ V, xj ∈ v
⊂ T(I)

SLIDE 21

BUCHBERGER THEORY FOR EFFECTIVE ASSOCIATIVE RINGS

Zacharias G., Generalized Gr¨

bner bases in commutative

polynomial rings, Bachelor’s thesis, M.I.T. (1978) Effective description of the canonical form of the ring

SLIDE 22

BUCHBERGER THEORY FOR EFFECTIVE ASSOCIATIVE RINGS

Zacharias G., Generalized Gr¨

bner bases in commutative

polynomial rings, Bachelor’s thesis, M.I.T. (1978) Spear D.A., A constructive approach to commutative ring theory, in Proc. of the 1977 MACSYMA Users’ Conference, NASA CP-2012 (1977), 369–376 Importing a Buchberger Theory from ZZ to A

SLIDE 23

BUCHBERGER THEORY FOR EFFECTIVE ASSOCIATIVE RINGS

Zacharias G., Generalized Gr¨

bner bases in commutative

polynomial rings, Bachelor’s thesis, M.I.T. (1978) Spear D.A., A constructive approach to commutative ring theory, in Proc. of the 1977 MACSYMA Users’ Conference, NASA CP-2012 (1977), 369–376 M¨

ller H.M., On the construction of Gr¨
bner bases using

syzygies, J. Symb. Comp. 6 (1988), 345–359 Pritchard F . L., The ideal membership problem in non-commutative polynomial rings,

J. Symb. Comp. 22 (1996), 27–48

Lifting Theorem

SLIDE 24

ZACHARIAS CANONICAL FORM

The obvious canonical forms of Ac for Zc namely Ac := {r : 0 ≤ r < c} or Ac := {r : 0 < r ≤ c} or Ac := {r : − c

2 < r ≤ c 2} give naturally a computational canonical

form for ZZm For a module I ⊂ ZZm, and each term τ ∈ Z(m) =

υei, υ ∈ Z, 1 ≤ i ≤ m
,

considering the principal ideals I(cτ)u := {lc(f ) : f ∈ I, T(f ) = τ} ∪ {0} ⊂ Z, we have ZZm/I ∼ = Zach(ZZm/I) =:

τ∈Z(m)

Acτ τ I = I(4X, 2X 2) A = Z ⊕ Z4X ⊕ X 2Z2[X]

SLIDE 25

ZACHARIAS CANONICAL FORM

The obvious canonical forms of Ac for Zc namely Ac := {r : 0 ≤ r < c} or Ac := {r : 0 < r ≤ c} or Ac := {r : − c

2 < r ≤ c 2} give naturally a computational canonical

form for ZZm For a module I ⊂ ZZm, and each term τ ∈ Z(m) =

υei, υ ∈ Z, 1 ≤ i ≤ m
,

considering the principal ideals I(cτ)u := {lc(f ) : f ∈ I, T(f ) = τ} ∪ {0} ⊂ Z, we have ZZm/I ∼ = Zach(ZZm/I) =:

τ∈Z(m)

Acτ τ I = I(4X, 2X 2) A = Z ⊕ Z4X ⊕ X 2Z2[X]

SLIDE 26

ZACHARIAS CANONICAL FORM

Π : Q := Zx1, . . . , xj, . . . , X1, . . . , Xi, . . . ։ A = Q/I π : R = Zx1, . . . , xj, . . . ։ R = R/I

Xixj for each Xi ∈ V, xj ∈ v
⊂ T(I)

B := {ω ∈ V : Iω = R} ⊂

υω : υ ∈ v, ω ∈ V
A ∼

=

ω∈V

 

υ∈v

Acυωυ   ω =:

ω∈V

Rωω ⊂ RV ⊂ Q Rω := R/Iω ∼ =

υ∈v

Acυωυ ⊂ RV

SLIDE 27

ZACHARIAS CANONICAL FORM

Π : Q := Zx1, . . . , xj, . . . , X1, . . . , Xi, . . . ։ A = Q/I π : R = Zx1, . . . , xj, . . . ։ R = R/I

Xixj for each Xi ∈ V, xj ∈ v
⊂ T(I)

B := {ω ∈ V : Iω = R} ⊂

υω : υ ∈ v, ω ∈ V
A ∼

=

ω∈V

 

υ∈v

Acυωυ   ω =:

ω∈V

Rωω ⊂ RV ⊂ Q Rω := R/Iω ∼ =

υ∈v

Acυωυ ⊂ RV

SLIDE 28

ZACHARIAS CANONICAL FORM

Π : Q := Zx1, . . . , xj, . . . , X1, . . . , Xi, . . . ։ A = Q/I π : R = Zx1, . . . , xj, . . . ։ R = R/I

Xixj for each Xi ∈ V, xj ∈ v
⊂ T(I)

B := {ω ∈ V : Iω = R} ⊂

υω : υ ∈ v, ω ∈ V
A ∼

=

ω∈V

 

υ∈v

Acυωυ   ω =:

ω∈V

Rωω ⊂ RV ⊂ Q Rω := R/Iω ∼ =

υ∈v

Acυωυ ⊂ RV

SLIDE 29

ZACHARIAS CANONICAL FORM

Π : Q := Zx1, . . . , xj, . . . , X1, . . . , Xi, . . . ։ A = Q/I π : R = Zx1, . . . , xj, . . . ։ R = R/I

Xixj for each Xi ∈ V, xj ∈ v
⊂ T(I)

B := {ω ∈ V : Iω = R} ⊂

υω : υ ∈ v, ω ∈ V
A ∼

=

ω∈V

 

υ∈v

Acυωυ   ω =:

ω∈V

Rωω ⊂ RV ⊂ Q Rω := R/Iω ∼ =

υ∈v

Acυωυ ⊂ RV

SLIDE 30

SPEAR’S THEOREM

B := {ω ∈ V : Iω = R} ⊂

υω : υ ∈ v, ω ∈ V
Spear’s intuition that a Buchberger Theory defined in a ring can

be exported to its quotients allow us to impose on A the “natural” Γ-valuation/filtration T(·) : Am → B(m) : f → T(f ) where (Γ, ◦), B ⊂ Γ ⊂ V, is a suibale semigroup.

SLIDE 31

SPEAR’S THEOREM

B := {ω ∈ V : Iω = R} ⊂

υω : υ ∈ v, ω ∈ V
T(·) : Am → B(m) : f → T(f )

(Γ, ◦), B ⊂ Γ ⊂ V,

SLIDE 32

SPEAR’S THEOREM

B := {ω ∈ V : Iω = R} ⊂

υω : υ ∈ v, ω ∈ V
T(·) : Am → B(m) : f → T(f )

(Γ, ◦), B ⊂ Γ ⊂ V, The associated Γ-graded ring G = G(A) coincides as a set with A and this is sufficient to smoothly export Buchberger test/completion but they don’t conside as rings: the multiplication ⋆ of A does not coincide with the one, ∗, of G

SLIDE 33

SPEAR’S THEOREM

B := {ω ∈ V : Iω = R} ⊂

υω : υ ∈ v, ω ∈ V
T(·) : Am → B(m) : f → T(f )

(Γ, ◦), B ⊂ Γ ⊂ V, The associated Γ-graded ring G = G(A) coincides as a set with A and this is sufficient to smoothly export Buchberger test/completion but they don’t conside as rings: the multiplication ⋆ of A does not coincide with the one, ∗, of G For instance, if we consider the Weyl algebra, A = QD, X/I(DX − XD − 1) where G = Q[D, X], D ⋆ X = XD − 1, D ∗ X = XD.

SLIDE 34

SPEAR’S THEOREM

B := {ω ∈ V : Iω = R} ⊂

υω : υ ∈ v, ω ∈ V
T(·) : Am → B(m) : f → T(f )

(Γ, ◦), B ⊂ Γ ⊂ V, The associated Γ-graded ring G = G(A) coincides as a set with A and this is sufficient to smoothly export Buchberger test/completion but they don’t conside as rings: the multiplication ⋆ of A does not coincide with the one, ∗, of G However an old slogan stated that in order to provide a Buchberger Algorithm on A, one just needs to modify, in the algorithm for G, the multiplication procedure!

SLIDE 35

GR ¨

OBNER BASES

f =

s

i=1

c(f , ti)ti : c(f , ti) ∈ Z \ {0}, ti ∈ Z, t1 > · · · > ts. T(f ) := t1, lc(f ) := c(f , t1), M(f ) := c(f , t1)t1. Pan: XY = 3X · Y − X · 2Y ∈ I(3X, 2Y ) Let M ⊂ Am be a (left, right, bilateral) A-module. F ⊂ M will be called

SLIDE 36

GR ¨

OBNER BASES

f =

s

i=1

c(f , ti)ti : c(f , ti) ∈ Z \ {0}, ti ∈ Z, t1 > · · · > ts. T(f ) := t1, lc(f ) := c(f , t1), M(f ) := c(f , t1)t1. Pan: XY = 3X · Y − X · 2Y ∈ I(3X, 2Y ) Let M ⊂ Am be a (left, right, bilateral) A-module. F ⊂ M will be called

SLIDE 37

GR ¨

OBNER BASES

f =

s

i=1

c(f , ti)ti : c(f , ti) ∈ Z \ {0}, ti ∈ Z, t1 > · · · > ts. T(f ) := t1, lc(f ) := c(f , t1), M(f ) := c(f , t1)t1. Pan: XY = 3X · Y − X · 2Y ∈ I(3X, 2Y ) Let M ⊂ Am be a (left, right, bilateral) A-module. F ⊂ M will be called

SLIDE 38

GR ¨

OBNER BASES

f =

s

i=1

c(f , ti)ti : c(f , ti) ∈ Z \ {0}, ti ∈ Z, t1 > · · · > ts. T(f ) := t1, lc(f ) := c(f , t1), M(f ) := c(f , t1)t1. Pan: XY = 3X · Y − X · 2Y ∈ I(3X, 2Y ) Let M ⊂ Am be a (left, right, bilateral) A-module. F ⊂ M will be called a (left, right, bilateral) Gr¨

bner basis of M if F satisfies the following

condition:

for each f ∈ M, there are gi ∈ F,

λi, ρi ∈ B, ai ∈ Rλi \ {0}, bi ∈ Rρi \ {0} such that

T(f ) = λi ◦ T(gi) ◦ ρi for all i,
M(f ) =

i aiλi ∗ M(gi) ∗ biρi;

SLIDE 39

GR ¨

OBNER BASES

f =

s

i=1

c(f , ti)ti : c(f , ti) ∈ Z \ {0}, ti ∈ Z, t1 > · · · > ts. T(f ) := t1, lc(f ) := c(f , t1), M(f ) := c(f , t1)t1. Pan: XY = 3X · Y − X · 2Y ∈ I(3X, 2Y ) Let M ⊂ Am be a (left, right, bilateral) A-module. F ⊂ M will be called a (left, right, bilateral) strong Gr¨

bner basis of I if F satisfies the

following equivalent conditions

for each f ∈ M there is g ∈ F such that M(g) | M(f ),
for each f ∈ M there are g ∈ F,

λ, ρ ∈ B, a ∈ Rλ \ {0}, b ∈ Rρ \ {0}, such that T(f ) = λ ◦ T(g) ◦ ρ and M(f ) = aλ ∗ M(g) ∗ bρ.

SLIDE 40

BUCHBERGER THEOREM

THEOREM For any set F ⊂ Am \ {0} the following conditions are equivalent:

f ∈ I(F) ⇐

⇒ it has a representation f =

µ

i=1

aiλi ⋆ gi ⋆ biρi T(f ) = λ1 ◦T(g1)◦ρ1 and λi ◦T(gi)◦ρi > λi+1 ◦T(gi+1)◦ρi+1∀i;

F is a strong Gr¨
bner basis of I(F);

SLIDE 41

BUCHBERGER THEOREM

THEOREM For any set F ⊂ Am \ {0} the following conditions are equivalent:

f ∈ I(F) ⇐

⇒ it has a representation f =

µ

i=1

aiλi ⋆ gi ⋆ biρi T(f ) = λ1 ◦ T(g1) ◦ ρ1 and λi ◦ T(gi) ◦ ρi≥ λi+1 ◦ T(gi+1) ◦ ρi+1∀i;

F is a weak Gr¨
bner basis of I(F);

SLIDE 42

BUCHBERGER REDUCTION

In order to perform Buchberger Algorithm we need to solve PROBLEM Given g ∈ Am \ {0} and F ⊂ Am \ {0} decide whether M(g) ∈ M{I2(M{F})} ⊂ Gm in which case return gi ∈ F, λi, ρi ∈ B, ai ∈ Rλi \ {0}, bi ∈ Rρi \ {0} such that

T(g) = λi ◦ T(gi) ◦ ρi for all i and
M(g) =

i aiλi ∗ M(gi) ∗ biρi.

which is trivial in Q. Then in G performs g → g −

i aiλi ⋆ gi ⋆ biρi

SLIDE 43

LIFTING THEOREM

Given a finite set F := {g1, . . . , gu} ⊂ Am, gi = M(gi) − pi =: aiτieιi − pi, we denote M := I(F) and the morphisms sL : Gu → Gm and SL : Au → Am defined as sL u

i=1
ω∈B

aiωω

ei
:=

u

i=1
ω∈B

aiωω ∗ M(gi), SL u

i=1
ω∈B

aiωω

ei
:=

u

i=1
ω∈B

aiωω ⋆ gi,

SLIDE 44

LIFTING THEOREM

sL u

i=1
ω∈B

aiωω

ei
:=

u

i=1
ω∈B

aiωω ∗ M(gi), SL u

i=1
ω∈B

aiωω

ei
:=

u

i=1
ω∈B

aiωω ⋆ gi, A reformulation (and a more efficient procedure) of the classical Buchberger test-completion, which states that “a basis F is Gr¨

bner

if and only if each S-polynomial reduces to 0”.

SLIDE 45

LIFTING THEOREM

sL u

i=1
ω∈B

aiωω

ei
:=

u

i=1
ω∈B

aiωω ∗ M(gi), SL u

i=1
ω∈B

aiωω

ei
:=

u

i=1
ω∈B

aiωω ⋆ gi, A basis F is Gr¨

bner if and only if each element u in a minimal basis
f the module ker(s) of the syzygies among the leading monomials

M(gi) lifts, via Buchberger reduction of S(u), to a syzygy U ∈ ker(S) among the gi.

SLIDE 46

LIFTING THEOREM

sL u

i=1
ω∈B

aiωω

ei
:=

u

i=1
ω∈B

aiωω ∗ M(gi), SL u

i=1
ω∈B

aiωω

ei
:=

u

i=1
ω∈B

aiωω ⋆ gi, A basis F is Gr¨

bner if and only if each element u in a minimal basis
f the module ker(s) of the syzygies among the leading monomials

M(gi) lifts, via Buchberger reduction of S(u), to a syzygy U ∈ ker(S) among the gi.More precisely we have U = S(u) −

µ

i=1

aiλi ⋆ ei ⋆ biρi where µ

i=1 aiλi ⋆ gi ⋆ biρi is the Gr¨

bner repr. of S(u) mod. F.

SLIDE 47

LIFTING THEOREM

sL u

i=1
ω∈B

aiωω

ei
:=

u

i=1
ω∈B

aiωω ∗ M(gi), SL u

i=1
ω∈B

aiωω

ei
:=

u

i=1
ω∈B

aiωω ⋆ gi, A basis F is Gr¨

bner if and only if each element u in a minimal basis
f the module ker(s) of the syzygies among the leading monomials

M(gi) lifts, via Buchberger reduction of S(u), to a syzygy U ∈ ker(S) among the gi. As a corollary you get Janet–Schreier Theorem that the lifted ele- ments form a Gr¨

bner basis of ker(S)