Special Semester Presentation · Semester Presentation Gareth Evans Outline Commutative vs....

SpecialSemester

Presentation

Gareth Evans

Outline

Commutativevs. Noncom-mutative

What is anInvolutiveBasis?

Noncomm.InvolutiveBases

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

Special Semester PresentationNoncommutative Involutive Bases / Noncommutative

Grobner Walks

Gareth Evans

Workshop B2February / March, 2006

SpecialSemester

Presentation

Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

1 Commutative vs. Noncommutative

2 What is an Involutive Basis?

3 Noncomm. Involutive BasesDivisorsAlgorithmPropertiesDivisionsOpen Questions

4 WalksCommutative WalksNoncommutative Walks

SpecialSemester

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Gareth Evans

Outline

Divisors

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CommutativeWalks

NoncommutativeWalks

Commutative vs. Noncommutative

• In the commutative case, there is one S-polynomial forevery pair of polynomials.

• In the noncommutative case, the number of S-polynomialsper pair of polynomials is determined by the overlapsbetween the lead monomials of the polynomials.

• The commutative algorithm (Buchberger’s algorithm)always terminates; the noncommutative algorithm (Mora’salgorithm) does not.

SpecialSemester

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Gareth Evans

Outline

Divisors

Algorithm

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CommutativeWalks

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SpecialSemester

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Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

SpecialSemester

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Gareth Evans

Outline

Divisors

Algorithm

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Open Questions

CommutativeWalks

NoncommutativeWalks

What is an Involutive Basis?

• An Involutive Basis is a Grobner Basis such that uniqueremainders are also obtained uniquely.

• Not all conventional divisors are involutive divisors.

• An Involutive Basis is computed by working withprolongations and the process of autoreduction.

• Prolongation: a multiple of a basis polynomial by anonmultiplicative variable.

• Autoreduction: keeps the basis ‘minimised’ at all times.

• An Involutive division is chosen to decide which variablesare multiplicative for a particular polynomial p in a set ofpolynomials P.

• Popular choices of involutive division include the Thomas,Pommaret and Janet divisions.

• The Involutive Basis algorithm is guaranteed to terminateif the involutive division used satisfies certain properties.

SpecialSemester

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Gareth Evans

Outline

Divisors

Algorithm

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Open Questions

CommutativeWalks

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SpecialSemester

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Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

SpecialSemester

Presentation

Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

SpecialSemester

Presentation

Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

SpecialSemester

Presentation

Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

SpecialSemester

Presentation

Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

SpecialSemester

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Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

SpecialSemester

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Gareth Evans

Outline

Divisors

Algorithm

Properties

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Open Questions

CommutativeWalks

NoncommutativeWalks

SpecialSemester

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CommutativeWalks

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Definition (Thomas)

Let U = {u1, . . . , um} be a set of monomials over a polynomialring R[x1, . . . , xn], where the monomial uj ∈ U (for 1 6 j 6 m)has corresponding multidegree (e1

j , e2j , . . . , en

The Thomas involutive division T assigns multiplicativevariables to elements of U as follows: the variable xi ismultiplicative for monomial uj (written xi ∈MT (uj ,U)) ife ij = maxk e i

k for all 1 6 k 6 m.

Example

Monomial x5y2z y2z x2y2z xyz3 xz3 x4yz2 z

Thomas {x , y} {y} {y} {z} {z} ∅ ∅

SpecialSemester

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Definition (Thomas)

j , e2j , . . . , en

j ).The Thomas involutive division T assigns multiplicativevariables to elements of U as follows: the variable xi ismultiplicative for monomial uj (written xi ∈MT (uj ,U)) ife ij = maxk e i

Example

Thomas {x , y} {y} {y} {z} {z} ∅ ∅

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Definition (Thomas)

j , e2j , . . . , en

j ).The Thomas involutive division T assigns multiplicativevariables to elements of U as follows: the variable xi ismultiplicative for monomial uj (written xi ∈MT (uj ,U)) ife ij = maxk e i

Example

Thomas {x , y} {y} {y} {z} {z} ∅ ∅

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Consider the Janet Involutive BasisH := {xy − z , yz + 2x + z , 2x2 + xz + z2, 2x2z + xz2 + z3}and the corresponding Grobner BasisG := {xy − z , yz + 2x + z , 2x2 + xz + z2}.

Grobner Basis Involutive Basis

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CommutativeWalks

NoncommutativeWalks

Consider the Janet Involutive BasisH := {xy − z , yz + 2x + z , 2x2 + xz + z2, 2x2z + xz2 + z3}and the corresponding Grobner BasisG := {xy − z , yz + 2x + z , 2x2 + xz + z2}.

Grobner Basis Involutive Basis

SpecialSemester

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Gareth Evans

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CommutativeWalks

NoncommutativeWalks

• The Involutive Basis Algorithm can be thought of as analternative to Buchberger’s Algorithm.

• Which is more efficient?

• An Involutive Basis has extra combinatorial properties, e.g.simple deduction of the Hilbert function.

• More information:Calmet, Hausdorf and Seiler:

A Constructive Introduction to Involution;Gerdt and Blinkov:

Involutive Bases of Polynomial Ideals.

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SpecialSemester

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Gareth Evans

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CommutativeWalks

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SpecialSemester

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Gareth Evans

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CommutativeWalks

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CommutativeWalks

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Noncommutative Involutive Bases

• Need left/right multiplicative variables.

• When is a conventional divisor an involutive divisor?• Thin divisor:

• Thick divisor:

X X X X X

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• Thick divisor:

X X X X X

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• When is a conventional divisor an involutive divisor?

• Thin divisor:

• Thick divisor:

X X X X X

SpecialSemester

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Gareth Evans

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CommutativeWalks

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• Thick divisor:

X X X X X

SpecialSemester

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Gareth Evans

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Algorithm

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CommutativeWalks

NoncommutativeWalks

• Thick divisor:

X X X X X

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Input: A Basis F = {f1, f2, . . . , fm} for an ideal J over a noncommutativepolynomial ring R〈x1, . . . xn〉; an admissible monomial ordering O; an involutivedivision I .

Output: A Locally Involutive Basis G = {g1, g2, . . . , gp} for J (in the case oftermination).

G = ∅;F = Autoreduce(F );while (G == ∅) do

S = {xi f | f ∈ F , xi /∈ MLI (f , F )} ∪ {fxi | f ∈ F , xi /∈MR

I (f , F )};s′ = 0;while (S 6= ∅) and (s′ == 0) do

Let s be a polynomial in S whose lead monomial is minimal withrespect to O;S = S \ {s};s′ = RemI (s, F );

end whileif (s′ 6= 0) then

F = Autoreduce(F ∪ {s′});else

G = F ;end if

end whilereturn G ;

SpecialSemester

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Example

Let F := {f1, f2} = {x2y2 − 2xy2 + x2, x2y − 2xy} be a basisfor an ideal J over the polynomial ring Q〈x , y〉, and let themonomial ordering be DegLex.

Assume multiplicative variables for F as follows.

Polynomial MLI (fi ,F ) MR

I (fi ,F )

f1 = x2y2 − 2xy2 + x2 {x , y} {y}f2 = x2y − 2xy {x , y} {x}

Autoreduction does not alter the set, so we construct the set ofprolongationsS = {f1x , f2y} = {x2y2x − 2xy2x + x3, x2y2 − 2xy2}.

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Example

Let F := {f1, f2} = {x2y2 − 2xy2 + x2, x2y − 2xy} be a basisfor an ideal J over the polynomial ring Q〈x , y〉, and let themonomial ordering be DegLex.Assume multiplicative variables for F as follows.

I (fi ,F )

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Example

Let F := {f1, f2} = {x2y2 − 2xy2 + x2, x2y − 2xy} be a basisfor an ideal J over the polynomial ring Q〈x , y〉, and let themonomial ordering be DegLex.Assume multiplicative variables for F as follows.

I (fi ,F )

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As x2y2 < x2y2x in the DegLex monomial ordering, weinvolutively reduce the element f2y ∈ S first.

f2y = x2y2 − 2xy2

// x2y2 − 2xy2 − (x2y2 − 2xy2 + x2)

= −x2.

As the prolongation did not involutively reduce to zero, we nowexit from the second while loop of the algorithm and proceedby autoreducing the setF ∪ {f3 := −x2} = {x2y2 − 2xy2 + x2, x2y − 2xy , −x2}.(This of course requires a new assignment of multiplicativevariables; the algorithm eventually terminates with the setG = {−x2, −2xy , −2xy2, −2xyx , −2xy2x} as output.)

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f2y = x2y2 − 2xy2

// x2y2 − 2xy2 − (x2y2 − 2xy2 + x2)

= −x2.

As the prolongation did not involutively reduce to zero, we nowexit from the second while loop of the algorithm and proceedby autoreducing the setF ∪ {f3 := −x2} = {x2y2 − 2xy2 + x2, x2y − 2xy , −x2}.

(This of course requires a new assignment of multiplicativevariables; the algorithm eventually terminates with the setG = {−x2, −2xy , −2xy2, −2xyx , −2xy2x} as output.)

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f2y = x2y2 − 2xy2

// x2y2 − 2xy2 − (x2y2 − 2xy2 + x2)

= −x2.

As the prolongation did not involutively reduce to zero, we nowexit from the second while loop of the algorithm and proceedby autoreducing the setF ∪ {f3 := −x2} = {x2y2 − 2xy2 + x2, x2y − 2xy , −x2}.(This of course requires a new assignment of multiplicativevariables; the algorithm eventually terminates with the setG = {−x2, −2xy , −2xy2, −2xyx , −2xy2x} as output.)

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Input set F

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Input set F

apply algorithm��

Locally Involutive Basis

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Input set F

continuity��

Involutive Basis

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Input set F

continuity��

Involutive Basis

strong��

Grobner Basis

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Input set F

continuity��

Involutive Basis

strong��

Grobner Basis��

Grobner

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Definition (The Left Division)

Given any monomial u, the left division � assigns no leftnonmultiplicative variables to u, and assigns no rightmultiplicative variables to u (in other words, all variables areleft multiplicative and right nonmultiplicative for u).

RemarkThe Left Division is strong and continuous.

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Definition (The Left Division)

Given any monomial u, the left division � assigns no leftnonmultiplicative variables to u, and assigns no rightmultiplicative variables to u (in other words, all variables areleft multiplicative and right nonmultiplicative for u).

RemarkThe Left Division is strong and continuous.

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To illustrate the difference between the overlapping cones of anoncommutative Grobner Basis and the disjoint cones of anoncommutative Involutive Basis with respect to the leftdivision, consider the following example.

Example

Let F := {2xy + y2 + 5, x2 + y2 + 8} be a basis over thepolynomial ring Q〈x , y〉, and let the monomial ordering beDegLex.Applying Mora’s algorithm to F , we obtain the Grobner BasisG := {2xy +y2 +5, x2 +y2 +8, 5y3−10x +37y , 2yx +y2 +5}.Applying the noncommutative Involutive Basis algorithm to F(with respect to the left involutive division), we obtain theInvolutive Basis H := {2xy + y2 + 5, x2 + y2 + 8, 5y3 − 10x +37y , 5xy2 + 5x − 6y , 2yx + y2 + 5}.

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Example

Let F := {2xy + y2 + 5, x2 + y2 + 8} be a basis over thepolynomial ring Q〈x , y〉, and let the monomial ordering beDegLex.

Applying Mora’s algorithm to F , we obtain the Grobner BasisG := {2xy +y2 +5, x2 +y2 +8, 5y3−10x +37y , 2yx +y2 +5}.Applying the noncommutative Involutive Basis algorithm to F(with respect to the left involutive division), we obtain theInvolutive Basis H := {2xy + y2 + 5, x2 + y2 + 8, 5y3 − 10x +37y , 5xy2 + 5x − 6y , 2yx + y2 + 5}.

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Example

Let F := {2xy + y2 + 5, x2 + y2 + 8} be a basis over thepolynomial ring Q〈x , y〉, and let the monomial ordering beDegLex.Applying Mora’s algorithm to F , we obtain the Grobner BasisG := {2xy +y2 +5, x2 +y2 +8, 5y3−10x +37y , 2yx +y2 +5}.

Applying the noncommutative Involutive Basis algorithm to F(with respect to the left involutive division), we obtain theInvolutive Basis H := {2xy + y2 + 5, x2 + y2 + 8, 5y3 − 10x +37y , 5xy2 + 5x − 6y , 2yx + y2 + 5}.

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Example

Let F := {2xy + y2 + 5, x2 + y2 + 8} be a basis over thepolynomial ring Q〈x , y〉, and let the monomial ordering beDegLex.Applying Mora’s algorithm to F , we obtain the Grobner BasisG := {2xy +y2 +5, x2 +y2 +8, 5y3−10x +37y , 2yx +y2 +5}.Applying the noncommutative Involutive Basis algorithm to F(with respect to the left involutive division), we obtain theInvolutive Basis H := {2xy + y2 + 5, x2 + y2 + 8, 5y3 − 10x +37y , 5xy2 + 5x − 6y , 2yx + y2 + 5}.

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Noncommutative Involutive BasesGrobner Basis G = {2xy + y2 + 5, x2 + y2 + 8,5y3 − 10x + 37y , 2yx + y2 + 5}

/.-,()*+x2

�U�U

&f&f&f&f&f&f&f&f&f&f&f&f'&%$ !"#xy

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LLLLLLLLLLLLLLLLL '&%$ !"#yx

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r r r r r r r r r

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x4 x3y x2yx xyx2 yx3 x2y2 xy2x xyxy yxyx yx2y y2x2 xy3 yxy2 y2xy y3x y4

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Noncommutative Involutive BasesInvolutive Basis H = {2xy + y2 + 5, x2 + y2 + 8,5y3 − 10x + 37y , 5xy2 + 5x − 6y , 2yx + y2 + 5}

/.-,()*+x2

&f&f&f&f&f&f&f&f&f&f&f&f'&%$ !"#xy

LLLLLLLLLLLLLLLLL '&%$ !"#yx

r r r r r r r r r

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x4 x3y x2yx xyx2 yx3 x2y2 xy2x xyxy yxyx yx2y y2x2 xy3 yxy2 y2xy y3x y4

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Noncommutative Involutive BasesApplication: Complete Rewrite Systems for Groups.

Example

Let C := 〈Y ,X , y , x | x3 → ε, y2 → ε, (xy)2 → ε, Xx →ε, xX → ε, Yy → ε, yY → ε〉 be a monoid rws for S3.If we apply the Knuth-Bendix algorithm to C with respect tothe DegLex (word) ordering, we obtain the complete rewritesystem

C ′ := 〈Y ,X , y , x | xyx → y , yxy → X , x2 → X , Xx →ε, y2 → ε, Xy → yx , xX → ε, yX → xy , X 2 → x , Y → y〉.

The corresponding involutive complete rewrite system is

C ′′ := 〈Y ,X , y , x | y2 → ε, Xx → ε, xX → ε, Yy → ε, y2x →x , Y → y , Yx → yx , Xxy → y , Yyx → x , x2 → X , X 2 →

x , xyx → y , Xy → yx , Xyx → xy , x2y → yx , yX →xy , yxy → X , Yxy → X , YX → xy〉.

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Example

Let C := 〈Y ,X , y , x | x3 → ε, y2 → ε, (xy)2 → ε, Xx →ε, xX → ε, Yy → ε, yY → ε〉 be a monoid rws for S3.

If we apply the Knuth-Bendix algorithm to C with respect tothe DegLex (word) ordering, we obtain the complete rewritesystem

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Noncommutative Involutive BasesConsider the word yXYx . Using the 10 element completerewrite system C ′, there are several reduction paths for thisword, as illustrated by the following diagram.

yXYxyX→xy

yysssssss Y→y

%%KKKKKKK

Y→y %%KKKKKKK yXyx

yX→xyyysssssss Xy→yx

$$JJJJJJJ

y2→ε %%JJJJJJJJ y2x2

y2→ε

zzuuuuuuuu x2→X

$$IIIIII

x2→X %%KKKKKKKKK y2X

y2→εzztttttt

tt yX→xy

$$IIIIII

yxy→X��

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However, by involutively reducing the word yXYx with respectto the 19 element involutive complete rewrite system C ′′, thereis only one reduction path, namely

Yx→yx��

Xyx→xy

��yxy

yxy→X��

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Problem:With respect to the left division, the noncommutativeInvolutive Basis algorithm does not always terminate, given theexistence of a noncommutative Grobner Basis.

Solution:Try defining a different division!

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Problem:With respect to the left division, the noncommutativeInvolutive Basis algorithm does not always terminate, given theexistence of a noncommutative Grobner Basis.

Solution:Try defining a different division!

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Noncommutative Involutive BasesDefinition (The Left Overlap Division O)Let U = {u1, . . . , um} be a set of monomials, and assume that allvariables are left and right multiplicative for all elements of U tobegin with.

(a) For all possible ways that a monomial uj ∈ U is a subword of a(different) monomial ui ∈ U, so that

Subword(ui , k, k + deg(uj)− 1) = uj

for some integer k, if uj is not a suffix of ui , assign the variableSubword(ui , k + deg(uj), k + deg(uj)) to be rightnonmultiplicative for uj .

(b) For all possible ways that a proper prefix of a monomial ui ∈ U isequal to a proper suffix of a (not necessarily different) monomialuj ∈ U, so that

Prefix(ui , k) = Suffix(uj , k)

for some integer k and ui is not a subword of uj or vice-versa,assign the variable Subword(ui , k + 1, k + 1) to be rightnonmultiplicative for uj .

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Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

Example

Consider the set of polynomialsF := {xy − z , x + z , yz − z , xz , zy + z , z2}. Here are the leftand right multiplicative variables for LM(F ) with respect to theleft overlap division O.

u MLO(u, LM(F )) MR

O(u, LM(F ))

xy {x , y , z} {x , y}x {x , y , z} {x}yz {x , y , z} {x}xz {x , y , z} {x}zy {x , y , z} {x , y}z2 {x , y , z} {x}

SpecialSemester

Presentation

Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

Open Questions

• Are there any conclusive noncommutative involutivedivisions?

• Better algorithms?

• Applications?

More information:Evans: Noncommutative Involutive Bases (PhD Thesis,University of Wales, Bangor, 2005). Available on the arXiv.

SpecialSemester

Presentation

Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

Open Questions

• Applications?

SpecialSemester

Presentation

Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

Open Questions

• Applications?

SpecialSemester

Presentation

Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

Open Questions

• Applications?

SpecialSemester

Presentation

Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

Open Questions

• Applications?

SpecialSemester

Presentation

Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

Commutative Grobner andInvolutive Walks

• The ‘walk’ converts a Grobner or Involutive Basis withrespect to one monomial ordering to a Grobner orInvolutive Basis with respect to another monomialordering.

• It works with the matrices associated to the source andtarget monomial orderings. Example:

DegLex =

1 1 11 0 00 1 0

, Lex =

1 0 00 1 00 0 1

• The walk takes place on the line segment between the firsttwo rows of the source and target matrices.

• Each step of the walk computes a Grobner or InvolutiveBasis for a set of ‘initials’, determined by the first row ofthe current matrix.

SpecialSemester

Presentation

Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

DegLex =

1 1 11 0 00 1 0

, Lex =

1 0 00 1 00 0 1

SpecialSemester

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Gareth Evans

Outline

Divisors

Algorithm

Properties

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CommutativeWalks

NoncommutativeWalks

• It works with the matrices associated to the source andtarget monomial orderings.

Example:

DegLex =

1 1 11 0 00 1 0

, Lex =

1 0 00 1 00 0 1

SpecialSemester

Presentation

Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

DegLex =

1 1 11 0 00 1 0

, Lex =

1 0 00 1 00 0 1

SpecialSemester

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Gareth Evans

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Algorithm

Properties

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NoncommutativeWalks

DegLex =

1 1 11 0 00 1 0

, Lex =

1 0 00 1 00 0 1

SpecialSemester

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Gareth Evans

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Algorithm

Properties

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NoncommutativeWalks

DegLex =

1 1 11 0 00 1 0

, Lex =

1 0 00 1 00 0 1

SpecialSemester

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Gareth Evans

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Algorithm

Properties

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Open Questions

CommutativeWalks

NoncommutativeWalks

Example

Input: {xy − z , yz + 2x + z , 2x2 + xz + z2}, DegLex.

Output: {x + 12yz + 1

2z , y2z + yz + 2z}, Lex.

(1, 1, 1)Pass 1

Pass 3(1, 0, 0)

Pass 2(1, 1

SpecialSemester

Presentation

Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

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CommutativeWalks

NoncommutativeWalks

Example

Input: {xy − z , yz + 2x + z , 2x2 + xz + z2}, DegLex.Output: {x + 1

2yz + 12z , y2z + yz + 2z}, Lex.

(1, 1, 1)Pass 1

Pass 3(1, 0, 0)

Pass 2(1, 1

SpecialSemester

Presentation

Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

Example

Input: {xy − z , yz + 2x + z , 2x2 + xz + z2}, DegLex.Output: {x + 1

2yz + 12z , y2z + yz + 2z}, Lex.

(1, 1, 1)Pass 1

Pass 3(1, 0, 0)

Pass 2(1, 1

SpecialSemester

Presentation

Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

Noncommutative Walks

• Only a partial generalisation: Not allowed to walk betweenany two monomial orderings, only ‘harmonious’ ones.

Commutative Noncommutative

Matrices Functional DecompositionsRows Ordering Functions

DefinitionThe functional decomposition Θ = {θ1, θ2, . . .} correspondingto the DegLex monomial ordering is defined (for an arbitrarymonomial m) as follows.

θi (m) =

{deg(m) if i = 1.

n + 1− vali−1(m) if i > 1.

SpecialSemester

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Gareth Evans

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Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

θi (m) =

{deg(m) if i = 1.

n + 1− vali−1(m) if i > 1.

SpecialSemester

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Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

θi (m) =

{deg(m) if i = 1.

n + 1− vali−1(m) if i > 1.

SpecialSemester

Presentation

Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

θi (m) =

{deg(m) if i = 1.

n + 1− vali−1(m) if i > 1.

SpecialSemester

Presentation

Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

Results

• The basis of initials is a Grobner (or Involutive) Basis.

• The ‘lifted’ basis is a Grobner (or Involutive) Basis.

• Walks between harmonious monomial orderings, where thefirst ordering functions must be extendible and identical.

Problems

• How to find the next step on the walk?

• In particular, how to define an intermediate monomialordering that is admissible.

SpecialSemester

Presentation

Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

Results

Problems

SpecialSemester

Presentation

Gareth Evans

Outline

Divisors

Algorithm

Properties

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CommutativeWalks

NoncommutativeWalks

Results

Problems

SpecialSemester

Presentation

Gareth Evans

Outline

Divisors

Algorithm

Properties

Divisions

Open Questions

CommutativeWalks

NoncommutativeWalks

Results

Problems

SpecialSemester

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Gareth Evans

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Algorithm

Properties

Divisions

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CommutativeWalks

NoncommutativeWalks

Results

Problems

SpecialSemester

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Gareth Evans

Outline

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Algorithm

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CommutativeWalks

NoncommutativeWalks

More information (Commutative):Amrhein, Gloor and Kuchlin:

On the Walk;Collart, Kalkbrener and Mall:

Converting Bases with the Grobner Walk;Golubitsky:

Involutive Grobner Walk.

More information (Noncommutative):Evans:

Noncommutative Involutive Bases (PhD Thesis,University of Wales, Bangor, 2005).Available on the arXiv.

Special Semester Presentation · Semester Presentation Gareth Evans Outline Commutative vs....

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