Linear Algebra over Polynomial Ringsmath.usask.ca/~bremner/research/publications/Slides-for...Linear...

Linear Algebra over Polynomial Rings

Murray Bremner

University of Saskatchewan, Canada

Trinity College Dublin, Thursday 29 October 2015

Introduction

The main question I will address in this talk is

How does the rank of a matrix A with entries in a ring ofpolynomials F[x1, . . . , xk ] depend on the parameters?

In the simplest case of no parameters (matrices over the field F),the question is answered by Gaussian elimination, which allows usto compute the row canonical form (RCF) of the matrix.

In the case of one parameter, the polynomial ring F[x1] is a PID,and Gaussian elimination combined with the Euclidean algorithmfor GCDs allows us to compute the Hermite normal form (HNF).

For two or more parameters, we need the useful fact that

rank(A) = r if and only if at least one r × r minor isnonzero but every (r + 1)× (r + 1) minor is zero.

Introduction

Minors of a fixed size r in a given polynomial matrix A generatedeterminantal ideals DIr of A in the polynomial ring F[x1, . . . , xk ].

To find the zero sets V (DIr ) of these ideals we compute theirGrobner bases and possibly Grobner bases for the radicals

√DIr .

Large determinants are difficult to compute, especially with morethan two parameters, since we cannot use Gaussian elimination.

This leads us to search for canonical or at least reduced forms ofmatrices to make the determinantal ideals easier to compute.

Canonical forms of matrices over F[x1, . . . , xk ] are very close toGrobner bases for submodules of free modules over F[x1, . . . , xk ].

√DIr .

Matrices over a field F

A is an m × n matrix over the field F

rowspace(A) = subspace of Fn generated by rows of A

rank(A) = dim(rowspace(A))

modules over a field are free (vector spaces with dimension)

perform Gaussian elimination, use elementary row operations

compute RCF (row canonical form) of matrix

rank is the number of nonzero rows in the RCF

colspace(A) = subspace of Fm generated by columns of A

rank(A) = dim(colspace(A))

simultaneous row-column reduction: Smith normal form

A is an m × n matrix over the field Frowspace(A) = subspace of Fn generated by rows of A

Matrices over a PID

A is an m × n matrix over the PID R, say R = F[x ]

rowmodule(A) = submodule of Rn generated by rows of A

structure theory for finitely generated R-modules =⇒rowmodule(A) = free ⊕ torsion, but rowmodule(A) ⊆ Rn, andRn is free R-module, so torsion = 0, rowmodule(A) = free.

rank(A) = freerank(rowmodule(A))

assuming R is a Euclidean domain, we have an algorithm:

perform Gaussian elimination using elementary row operationsto compute pivots, perform Euclidean algorithm for GCDsusing row operations to put GCD in pivot positionresult is the HNF (Hermite normal form) of matrixrank is the number of nonzero rows in the HNF

same works for rows and columns (Smith normal form)

Matrices over a PID

HNF algorithm in detail

Input: An m × n matrix A with entries in a Euclidean domain R.

1 Set H ← A and i ← 1 and j ← 1.

2 While i ≤ m and j ≤ n do:

If hkj = 0 for k = i , . . . ,m then (all entries 0 at/below pivot)Set j ← j + 1

elseWhile there is a nonzero entry (strictly) below the pivot:

1 Find k with i ≤ k ≤ m such that hkj has minimal degree(depending on R) among nonzero entries at/below pivot.

2 If i 6= k then interchange rows i and k.3 Normalize hij depending on R (e.g. monic for polynomials).4 Use add-multiple row operations to reduce entries below pivot

to their remainders modulo hij .

Use add-multiple row operations to reduce entries above pivotto their remainders modulo hij .Set i ← i + 1 and j ← j + 1.

3 Return H.

If hkj = 0 for k = i , . . . ,m then (all entries 0 at/below pivot)

Set j ← j + 1

3 Return H.

While there is a nonzero entry (strictly) below the pivot:1 Find k with i ≤ k ≤ m such that hkj has minimal degree

(depending on R) among nonzero entries at/below pivot.2 If i 6= k then interchange rows i and k.3 Normalize hij depending on R (e.g. monic for polynomials).4 Use add-multiple row operations to reduce entries below pivot

3 Return H.

Use add-multiple row operations to reduce entries above pivotto their remainders modulo hij .

Set i ← i + 1 and j ← j + 1.

3 Return H.

Matrices over F[x1, . . . , xk ], F field, k ≥ 2

“Linear algebra over rings is lots more fun than overfields.” (R. Wiegand, “What is . . . a syzygy?”, Notices ofthe American Mathematical Society, April 2006)

One reason it’s more fun is that we have to ask ourselves what wemean by the rank in this case, and there are different answers.

The simplest answer: The integral domain R = F[x1, . . . , xk ] iscontained in its field of quotients Q(R), the rational functions:

R = F[x1, . . . , xk ] ⊂ F(x1, . . . , xk) = Q(R).

We can regard a matrix over R as a matrix over Q(R) and applyGaussian elimination to find its rank over Q(R).

But when we divide by f ∈ R, we erase the information containedin the zeros of f , so the results will be not be valid in general.

R = F[x1, . . . , xk ] ⊂ F(x1, . . . , xk) = Q(R).

Rank and determinants

A more useful notion of the rank of a matrix over F[x1, . . . , xk ] isgiven by taking the following characterization of the rank in thefield case as the definition of the rank in the polynomial case.

Definition

Let A be an m × n matrix over the commutative (associative)unital ring R. For 1 ≤ r ≤ min(m, n), by a minor of rank r wemean the determinant of any r × r submatrix of A.

Theorem

Let A be an m× n matrix over the field F. Then the rank of A is rif and only if at least one minor of rank r is not 0, and every minorof A of rank r + 1 is 0.

Definition

Theorem

Definition

Theorem

Definition

Theorem

If we replace the field F by the polynomial ring F[x1, . . . , xk ], then

“at least one minor of rank r is not 0” becomes“the ideal generated by the minors of rank r is not {0}”, and

“every minor of A of rank r + 1 is 0” becomes“the ideal generated by the minors of rank r + 1 is {0}”.

Definition

Let A be an m × n matrix over the polynomial ring F[x1, . . . , xk ].For 0 ≤ r ≤ min(m, n), the ideal DIr (A) generated by the minorsof A of rank r is called the r -th determinantal ideal of A.

For r = 0, there is(m0

)= 1 minor of rank 0, and it is nonzero

when every 1× 1 minor (every entry) of A is zero (A = O).The 0× 0 minor of any matrix A is 1, so DI0(A) = F[x1, . . . , xk ].The 1st determinantal ideal DI1(A) is generated by the entries of A.

Definition

when every 1× 1 minor (every entry) of A is zero (A = O).

The 0× 0 minor of any matrix A is 1, so DI0(A) = F[x1, . . . , xk ].The 1st determinantal ideal DI1(A) is generated by the entries of A.

Definition

when every 1× 1 minor (every entry) of A is zero (A = O).The 0× 0 minor of any matrix A is 1, so DI0(A) = F[x1, . . . , xk ].

The 1st determinantal ideal DI1(A) is generated by the entries of A.

Definition

Over a field, vanishing (resp. non-vanishing) of a determinant isequivalent to linear dependence (resp. independence) of the rows.

Moreover, sets containing a dependent set must also be dependent,and subsets of an independent set must also be independent.

(Good question for beginning linear algebra students: Is the emptyset dependent or independent? Only one answer makes sense!)

Over a field, these facts imply that the rank of a matrix iswell-defined: as r increases, there is a unique point at which ther × r minors switch from being not all 0 to being all 0.

Over a polynomial ring, analogous results hold and imply that thedeterminantal ideals are weakly decreasing:

F[x1, . . . , xk ] = DI0(A) ⊇ DI1(A) ⊇ DI2(A) ⊇ · · · ⊇ DImin(m,n)(A).

Definition

Let I be an ideal in F[x1, . . . , xk ]. The zero set of I , denoted V (I ),is the set of all points in Fk which satisfy every polynomial f ∈ I :

V (I ) = { (a1, . . . , ak) ∈ Fk | f (a1, . . . , ak) = 0, ∀ f ∈ I }.

The special cases of A with rank < r (strictly less than) areobtained by substituting the values (a1, . . . , ak) in the zero setZ (DIr (A)) for the parameters x1, . . . , xk in A.

(Recall that rank < r means every r × r minor is 0.)

The special cases of A with rank = r correspond to the values

(a1, . . . , ak) ∈ Z (DIr+1(A)) \ Z (DIr (A)).

(Rank < r + 1, but not < r .)

Definition

V (I ) = { (a1, . . . , ak) ∈ Fk | f (a1, . . . , ak) = 0, ∀ f ∈ I }.

(a1, . . . , ak) ∈ Z (DIr+1(A)) \ Z (DIr (A)).

Definition

V (I ) = { (a1, . . . , ak) ∈ Fk | f (a1, . . . , ak) = 0, ∀ f ∈ I }.

(a1, . . . , ak) ∈ Z (DIr+1(A)) \ Z (DIr (A)).

Definition

V (I ) = { (a1, . . . , ak) ∈ Fk | f (a1, . . . , ak) = 0, ∀ f ∈ I }.

(a1, . . . , ak) ∈ Z (DIr+1(A)) \ Z (DIr (A)).

Definition

V (I ) = { (a1, . . . , ak) ∈ Fk | f (a1, . . . , ak) = 0, ∀ f ∈ I }.

(a1, . . . , ak) ∈ Z (DIr+1(A)) \ Z (DIr (A)).

The relation between ideals and their zero sets is order-reversing:

I ⊆ J ⇐⇒ V (I ) ⊇ V (J).

(A smaller set of equations produces a larger set of solutions.)

Applying this to determinantal ideals, we first note that for any A,

V (DI0(A)) = V (F[x1, . . . , xk ]) = ∅.

We have a weakly increasing sequence of algebraic varieties in Fk :

∅ = V (DI0(A)) ⊆ V (DI1(A)) ⊆ · · · ⊆ V (DImin(m,n)(A)).

If at any step we have equality, V (DIr (A)) = V (DIr+1(A)), thenthere are no solutions of rank r . Notation: Vr = V (DIr (A)).

I ⊆ J ⇐⇒ V (I ) ⊇ V (J).

V (DI0(A)) = V (F[x1, . . . , xk ]) = ∅.

I ⊆ J ⇐⇒ V (I ) ⊇ V (J).

V (DI0(A)) = V (F[x1, . . . , xk ]) = ∅.

I ⊆ J ⇐⇒ V (I ) ⊇ V (J).

V (DI0(A)) = V (F[x1, . . . , xk ]) = ∅.

I ⊆ J ⇐⇒ V (I ) ⊇ V (J).

V (DI0(A)) = V (F[x1, . . . , xk ]) = ∅.

Example 1

Consider this 4× 4 matrix A with entries in F[x , y ]:

0 x x 10 y 1 1x y y 10 x y 0

We have DI0 = F[x , y ] and V0 = ∅.Since 1 is an entry of A we have DI1(A) = F[x , y ] and V1 = ∅.The monic 2× 2 minors of A are

x , x − 1, y , y − 1, y − x , x2, yx ,yx − x , yx − x2, y2 − x , y2 − y , y2 − yx .

Hence 1 ∈ DI2(A) giving DI2(A) = F[x , y ] and V2 = ∅.

Example 1

0 x x 10 y 1 1x y y 10 x y 0

Example 1

0 x x 10 y 1 1x y y 10 x y 0

Example 1

0 x x 10 y 1 1x y y 10 x y 0

We have DI0 = F[x , y ] and V0 = ∅.

Since 1 is an entry of A we have DI1(A) = F[x , y ] and V1 = ∅.The monic 2× 2 minors of A are

Example 1

0 x x 10 y 1 1x y y 10 x y 0

We have DI0 = F[x , y ] and V0 = ∅.Since 1 is an entry of A we have DI1(A) = F[x , y ] and V1 = ∅.

The monic 2× 2 minors of A are

Example 1

0 x x 10 y 1 1x y y 10 x y 0

Example 1

0 x x 10 y 1 1x y y 10 x y 0

x2, x2 − x , yx , yx − x , yx − x2,y2 − yx − y + x , y2 − 2yx + x2, y2 − yx + x2 − x ,yx2 − x2, yx2 − x3, y2x − x2,

Hence a Grobner basis for DI3(A) is {x , y} and V3 = {(0, 0)}.The only 4× 4 minor is the determinant

y2x − yx2 + x3 − x2 = x(y2 − yx + x2 − x).

This has the solutions

{(0, y) | y ∈ F

(x , y) | x ∈ F, y = 12

√x(4− 3x)

Hence a Grobner basis for DI3(A) is {x , y} and V3 = {(0, 0)}.

The only 4× 4 minor is the determinant

y2x − yx2 + x3 − x2 = x(y2 − yx + x2 − x).

{(0, y) | y ∈ F

(x , y) | x ∈ F, y = 12

√x(4− 3x)

y2x − yx2 + x3 − x2 = x(y2 − yx + x2 − x).

{(0, y) | y ∈ F

(x , y) | x ∈ F, y = 12

√x(4− 3x)

y2x − yx2 + x3 − x2 = x(y2 − yx + x2 − x).

{(0, y) | y ∈ F

(x , y) | x ∈ F, y = 12

√x(4− 3x)

Conclusions for rank(A):

Since V1 = V2 = ∅, the matrix A never has rank 0 or 1.

Since V3 = {(0, 0)}, the rank is 2 if and only if x = y = 0.

The rank is 3 if and only if (x , y) 6= (0, 0) and (x , y) ∈ V4.

The rank is 4 if and only if (x , y) /∈ V4.

Full rank occurs on a Zariski dense subset of F2.

This example illustrates what we mean by finding the rank of amatrix with entries in a polynomial ring: finding explicitly how therank depends on the values of the parameters.

Over the field of rational functions F(x , y), the rank of A is 4,which is the maximal rank obtained from values of the parameters.This is usually called the generic rank of the matrix.

Over the field of rational functions F(x , y), the rank of A is 4,which is the maximal rank obtained from values of the parameters.

This is usually called the generic rank of the matrix.

Generators for determinantal ideals can be hard to compute.

We need to calculate arbitrarily large determinants, but since thematrix entries are polynomials in more than one parameter, . . .we can’t use Gaussian elimination.

A small example: an 8× 24 matrix over F[x1, . . . , xk ] with k ≥ 2.The last column contains the number of r × r minors:

) (24r

)1 8 24 1922 28 276 77283 56 2024 1133444 70 10626 7438205 56 42504 2380224

6 28 134596 37686887 8 346104 27688328 1 735471 735471

We need to calculate arbitrarily large determinants, but since thematrix entries are polynomials in more than one parameter, . . .

we can’t use Gaussian elimination.

) (24r

)1 8 24 1922 28 276 77283 56 2024 1133444 70 10626 7438205 56 42504 2380224

6 28 134596 37686887 8 346104 27688328 1 735471 735471

) (24r

)1 8 24 1922 28 276 77283 56 2024 1133444 70 10626 7438205 56 42504 2380224

6 28 134596 37686887 8 346104 27688328 1 735471 735471

) (24r

)1 8 24 1922 28 276 77283 56 2024 1133444 70 10626 7438205 56 42504 2380224

6 28 134596 37686887 8 346104 27688328 1 735471 735471

) (24r

)1 8 24 1922 28 276 77283 56 2024 1133444 70 10626 7438205 56 42504 2380224

6 28 134596 37686887 8 346104 27688328 1 735471 735471

For each r = 1, . . . ,min(m, n), after we have computed all theminors which generate the determinantal ideal DIr (A), we then:

compute the Grobner basis for DIr (A) with respect to somemonomial order, which may be difficult if the generating setconsists of millions of polynomials of high degrees, andsolve the system of polynomial equations (obtained by settingevery Grobner basis element to zero) to find the zero set ofDIr (A); at this point it may (or may not) be helpful to firstcompute a Grobner basis for the radical

√DIr (A).

Since there are so many minors, we want to reduce the size of thematrix as much as possible before computing the minors.

We first recall the Smith normal form over a field or a PID.

Remark

Henry J. S. Smith was born in Dublin in 1826. His paper onnormal forms is “On systems of linear indeterminate equations andcongruences”, Phil. Trans. R. Soc. Lond. 151 (1) (1861) 293–326.

compute the Grobner basis for DIr (A) with respect to somemonomial order, which may be difficult if the generating setconsists of millions of polynomials of high degrees, and

solve the system of polynomial equations (obtained by settingevery Grobner basis element to zero) to find the zero set ofDIr (A); at this point it may (or may not) be helpful to firstcompute a Grobner basis for the radical

√DIr (A).

Remark

√DIr (A).

Remark

√DIr (A).

Remark

√DIr (A).

Remark

√DIr (A).

Remark

Theorem

Let A be an m× n matrix over a field F, or a PID R. There exist:

invertible matrices U (m ×m) and V (n × n), and

an r × r diagonal matrix D where r = rank(A),

such that

[D 00 0

Moreover, writing D = diag(d1, . . . , dr ) we may assume thatdi | di+1 for i = 1, . . . , r − 1 and d1, . . . , dr are invariant up tomultiplication by units (so in the case of a field they are all 1).

Definition

The matrix UAV is the Smith normal form of the matrix A.

The Smith normal form can be computed using elementary row andcolumn operations: a “two-sided” version of Gaussian elimination.

Theorem

such that

[D 00 0

Definition

Theorem

such that

[D 00 0

Definition

Now consider a matrix A with entries in F[x1, . . . , xk ] for k ≥ 2:

Typically, many of the entries of A will be nonzero scalars.

Using elementary row and column operations, we move thesenonzero scalars into the upper left corner of the matrix:

We put the nonzero scalars on the main diagonal.We scale them to be leading 1s.We use these leading 1s with row and column operations toeliminate the nonzero elements below and to the right.

This creates the largest possible identity matrix I in the upperleft corner, and forces all the information of the original matrixA into a (much smaller) block B in the lower right corner.

rank(A) = rank(I ) + rank(B)

The size of I gives a lower bound on rank(A).

We need to compute the determinantal ideals only for B.

We put the nonzero scalars on the main diagonal.

We scale them to be leading 1s.We use these leading 1s with row and column operations toeliminate the nonzero elements below and to the right.

We put the nonzero scalars on the main diagonal.We scale them to be leading 1s.

We use these leading 1s with row and column operations toeliminate the nonzero elements below and to the right.

Definition

This process — using elementary row and column operations toreduce the original matrix A to an upper left identity block I and alower right block B with no nonzero scalar entries — is calledcomputing a partial Smith form of A.

If A has size m × n and entries in R = F[x1, . . . , xk ], then anypartial Smith form of A belongs to the orbit of A under the actionof the group Em(R)× En(R) where E stands for the groupgenerated by the elementary matrices:

A −→ UAV =

[I 00 B

], U ∈ Em(R), V ∈ En(R).

Note: GLn(R) is not necessarily equal to En(R) for general rings.

Definition

A −→ UAV =

[I 00 B

], U ∈ Em(R), V ∈ En(R).

Definition

A −→ UAV =

[I 00 B

], U ∈ Em(R), V ∈ En(R).

Definition

A −→ UAV =

[I 00 B

], U ∈ Em(R), V ∈ En(R).

Example 2

Consider this 8× 12 matrix over F[a, b], writing dot for zero tohighlight the nonzero entries:

1 a b . . . . . . . . .1 . . a b . . . . . . .. 1 . . . a . . b . . .. . 1 . . . a . . b . .. . . 1 . . . a . . b .. . . . 1 . . . a . . b. . . . . 1 a b . . . .. . . . . . . . . 1 a b

Every row has a leading 1; there are two leading 1s in column 1;there is a sequence of leading 1s just below the diagonal.

Example 2

A partial Smith form for this matrix is as follows:

1 . . . . . . . . . . .. 1 . . . . . . . . . .. . 1 . . . . . . . . .. . . 1 . . . . . . . .. . . . 1 . . . . . . .. . . . . 1 . . . . . .. . . . . . 1 . . . . .

. . . . . . . −a2b−a2 . −a3+ab −ab2−ab −b3−b2

The upper left identity block has size 7; the lower right block Bhas size 1× 5, with only four nonzero entries; in factored form:

−a2(b + 1), −a(a2 − b), −ab(b + 1), −b2(b + 1).

1 . . . . . . . . . . .. 1 . . . . . . . . . .. . 1 . . . . . . . . .. . . 1 . . . . . . . .. . . . 1 . . . . . . .. . . . . 1 . . . . . .. . . . . . 1 . . . . .

. . . . . . . −a2b−a2 . −a3+ab −ab2−ab −b3−b2

−a2(b + 1), −a(a2 − b), −ab(b + 1), −b2(b + 1).

1 . . . . . . . . . . .. 1 . . . . . . . . . .. . 1 . . . . . . . . .. . . 1 . . . . . . . .. . . . 1 . . . . . . .. . . . . 1 . . . . . .. . . . . . 1 . . . . .

. . . . . . . −a2b−a2 . −a3+ab −ab2−ab −b3−b2

−a2(b + 1), −a(a2 − b), −ab(b + 1), −b2(b + 1).

1 . . . . . . . . . . .. 1 . . . . . . . . . .. . 1 . . . . . . . . .. . . 1 . . . . . . . .. . . . 1 . . . . . . .. . . . . 1 . . . . . .. . . . . . 1 . . . . .

. . . . . . . −a2b−a2 . −a3+ab −ab2−ab −b3−b2

−a2(b + 1), −a(a2 − b), −ab(b + 1), −b2(b + 1).

The ideal DI1(B) ⊂ F[a, b] generated by these four nonzero entrieshas this pure lex Grobner basis (a ≺ b):

a2(a2 + 1), a(b − a2), b2(b + 1).

This is a zero-dimensional ideal, and its finite solution set is

(a, b, ) = (0, 0), (0,−1), (±i ,−1).

For these values of (a, b) the original matrix R has rank 7.

For all other pairs (a, b) ∈ F2 the rank of R is 8.

a2(a2 + 1), a(b − a2), b2(b + 1).

(a, b, ) = (0, 0), (0,−1), (±i ,−1).

a2(a2 + 1), a(b − a2), b2(b + 1).

(a, b, ) = (0, 0), (0,−1), (±i ,−1).

a2(a2 + 1), a(b − a2), b2(b + 1).

(a, b, ) = (0, 0), (0,−1), (±i ,−1).

a2(a2 + 1), a(b − a2), b2(b + 1).

(a, b, ) = (0, 0), (0,−1), (±i ,−1).

a2(a2 + 1), a(b − a2), b2(b + 1).

(a, b, ) = (0, 0), (0,−1), (±i ,−1).

Where did the last example come from?

Start with a ternary operation (−,−,−) satisfying this relation R:

((−,−,−),−,−) + a(−, (−,−,−),−) + b(−,−, (−,−,−)) ≡ 0.

In every monomial of arity (degree) n, the n dashes representthe n arguments x1, . . . , xn which always occur in the sameorder from left to right (which is why we can omit them).

In other words, only the identity permutation of the subscriptscan occur (this is what is known as a nonsymmetric operad).

That is, (−,−,−) = (x1, x2, x3) and

((−,−,−),−,−) = ((x1, x2, x3), x4, x5),

(−, (−,−,−),−) = (x1, (x2, x3, x4), x5),

(−,−, (−,−,−)) = (x1, x2, (x3, x4, x5)).

((−,−,−),−,−) + a(−, (−,−,−),−) + b(−,−, (−,−,−)) ≡ 0.

That is, (−,−,−) = (x1, x2, x3) and

((−,−,−),−,−) = ((x1, x2, x3), x4, x5),

(−, (−,−,−),−) = (x1, (x2, x3, x4), x5),

(−,−, (−,−,−)) = (x1, x2, (x3, x4, x5)).

((−,−,−),−,−) + a(−, (−,−,−),−) + b(−,−, (−,−,−)) ≡ 0.

That is, (−,−,−) = (x1, x2, x3) and

((−,−,−),−,−) = ((x1, x2, x3), x4, x5),

(−, (−,−,−),−) = (x1, (x2, x3, x4), x5),

(−,−, (−,−,−)) = (x1, x2, (x3, x4, x5)).

((−,−,−),−,−) + a(−, (−,−,−),−) + b(−,−, (−,−,−)) ≡ 0.

That is, (−,−,−) = (x1, x2, x3) and

((−,−,−),−,−) = ((x1, x2, x3), x4, x5),

(−, (−,−,−),−) = (x1, (x2, x3, x4), x5),

(−,−, (−,−,−)) = (x1, x2, (x3, x4, x5)).

((−,−,−),−,−) + a(−, (−,−,−),−) + b(−,−, (−,−,−)) ≡ 0.

That is, (−,−,−) = (x1, x2, x3) and

((−,−,−),−,−) = ((x1, x2, x3), x4, x5),

(−, (−,−,−),−) = (x1, (x2, x3, x4), x5),

(−,−, (−,−,−)) = (x1, x2, (x3, x4, x5)).

((−,−,−),−,−) + a(−, (−,−,−),−) + b(−,−, (−,−,−)) ≡ 0.

That is, (−,−,−) = (x1, x2, x3) and

((−,−,−),−,−) = ((x1, x2, x3), x4, x5),

(−, (−,−,−),−) = (x1, (x2, x3, x4), x5),

(−,−, (−,−,−)) = (x1, x2, (x3, x4, x5)).

Consider the consequences of R(−,−,−,−,−) obtained bysubstituting (−,−,−) for one argument of R, for example

R(−,−, (−,−,−),−,−),

or R for one argument of (−,−,−), for example

(−,R(−,−,−,−,−),−).

We obtain altogether 5 + 3 = 8 different relations in arity 7:

(((−−−)−−)−−) + a((−−−)(−−−)−) + b((−−−)−(−−−)),((−(−−−)−)−−) + a(−((−−−)−−)−) + b(−(−−−)(−−−)),((−−(−−−))−−) + a(−(−(−−−)−)−) + b(−−((−−−)−−)),((−−−)(−−−)−) + a(−(−−(−−−))−) + b(−−(−(−−−)−)),((−−−)−(−−−)) + a(−(−−−)(−−−)) + b(−−(−−(−−−))),(((−−−)−−)−−) + a((−(−−−)−)−−) + b((−−(−−−))−−),(−((−−−)−−)−) + a(−(−(−−−)−)−) + b(−(−−(−−−))−),(−−((−−−)−−)) + a(−−(−(−−−)−)) + b(−−(−−(−−−))).

R(−,−, (−,−,−),−,−),

(−,R(−,−,−,−,−),−).

R(−,−, (−,−,−),−,−),

(−,R(−,−,−,−,−),−).

R(−,−, (−,−,−),−,−),

(−,R(−,−,−,−,−),−).

There are 12 distinct ternary monomials in arity 7, which we canorder as follows, to verify that we have them all:

(((−−−)−−)−−), ((−(−−−)−)−−), ((−−(−−−))−−),

(−((−−−)−−)−), (−(−(−−−)−)−), (−(−−(−−−))−),

(−−((−−−)−−)), (−−(−(−−−)−)), (−−(−−(−−−))),

((−−−)(−−−)−), ((−−−)−(−−−)), (−(−−−)(−−−)).

Hence the space of relations in arity 7 which are consequences ofthe relation R in arity 5 is the row space of an 8× 12 matrix,which is the matrix considered in the last Example.

(((−−−)−−)−−), ((−(−−−)−)−−), ((−−(−−−))−−),

(−((−−−)−−)−), (−(−(−−−)−)−), (−(−−(−−−))−),

(−−((−−−)−−)), (−−(−(−−−)−)), (−−(−−(−−−))),

((−−−)(−−−)−), ((−−−)−(−−−)), (−(−−−)(−−−)).

(((−−−)−−)−−), ((−(−−−)−)−−), ((−−(−−−))−−),

(−((−−−)−−)−), (−(−(−−−)−)−), (−(−−(−−−))−),

(−−((−−−)−−)), (−−(−(−−−)−)), (−−(−−(−−−))),

((−−−)(−−−)−), ((−−−)−(−−−)), (−(−−−)(−−−)).

Example 3

For this, I’ll start with the motivation from algebraic operads.

The associativity relation (ab)c ≡ a(bc) implies that we canomit parentheses in every degree without causing ambiguity.

There are obvious and non-obvious analogues of associativityfor two operations; the best known of the latter are thediassociative relations for left and right operations a and `:

(a a b) a c ≡ a a (b a c), a a (b a c) ≡ a a (b ` c),

(a ` b) ` c ≡ a ` (b ` c), (a ` b) ` c ≡ (a a b) ` c ,

(a ` b) a c ≡ a ` (b a c).

On the left we have left, right, and inner associativity.

On the right we have the bar relations: on the bar side of theoperation symbols, the operation doesn’t matter.

Example 3

(a ` b) ` c ≡ a ` (b ` c), (a ` b) ` c ≡ (a a b) ` c ,

(a ` b) a c ≡ a ` (b a c).

Example 3

(a ` b) ` c ≡ a ` (b ` c), (a ` b) ` c ≡ (a a b) ` c ,

(a ` b) a c ≡ a ` (b a c).

Example 3

(a ` b) ` c ≡ a ` (b ` c), (a ` b) ` c ≡ (a a b) ` c ,

(a ` b) a c ≡ a ` (b a c).

Example 3

(a ` b) ` c ≡ a ` (b ` c), (a ` b) ` c ≡ (a a b) ` c ,

(a ` b) a c ≡ a ` (b a c).

Example 3

(a ` b) ` c ≡ a ` (b ` c), (a ` b) ` c ≡ (a a b) ` c ,

(a ` b) a c ≡ a ` (b a c).

Theorem

These relations imply that any monomial m of degree n in thevariables a1, . . . , an (from left to right) with any placement ofparentheses and any choice of operation symbols, has a uniquelydefined center ai such that m is equal to its normal form:

m = (a1 ` · · · ` ai−1) ` ai a (ai+1 a · · · a an).

Corollary

In the free diassociative algebra, there are n distinct normal formsin degree n for the monomial with the identity permutation of thevariables (just as in the free associative algebra there is only onedistinct normal form in every degree for the monomial with theidentity permutation of the variables).

Theorem

These relations imply that any monomial m of degree n in thevariables a1, . . . , an (from left to right) with any placement ofparentheses and any choice of operation symbols, has a uniquelydefined center ai such that m is equal to its normal form:

m = (a1 ` · · · ` ai−1) ` ai a (ai+1 a · · · a an).

Corollary

In the free diassociative algebra, there are n distinct normal formsin degree n for the monomial with the identity permutation of thevariables (just as in the free associative algebra there is only onedistinct normal form in every degree for the monomial with theidentity permutation of the variables).

Question

Are there any other sets of nonsymmetric relations in degree 3for two operations (associative or nonassociative) whichproduce exactly n normal forms in degree n for all n ≥ 1?

In other words, how special are the diassociative relations?

For the general case, we’ll use the operation symbols {•, ◦}.There is always one normal form in degree 1, namely a1.

There are always two normal forms in degree 2: a1 •a2, a1 ◦a2.

In degree 3, there are 8 distinct monomials: 2 placements ofparentheses, and 2 choices of operations in each of 2 positions:

(a1 • a2) • a3, (a1 • a2) ◦ a3, (a1 ◦ a2) • a3, (a1 ◦ a2) ◦ a3,a1 • (a2 • a3), a1 • (a2 ◦ a3), a1 ◦ (a2 • a3), a1 ◦ (a2 ◦ a3).

Question

For the general case, we’ll use the operation symbols {•, ◦}.

There is always one normal form in degree 1, namely a1.

Question

∃ 3 normal forms in degree 3 ⇐⇒ ∃ 5 independent relations.

Represent relations by coefficient vectors in monomial basis.

Span of relations is row space of unique 5× 8 matrix in RCF.

This is the relation matrix denoted R, and {R} ↔ Gr(8, 5).

Grassmannian Gr(8, 5) of 5-dim’l subspaces of 8-dim’l space.

Partition Gr(8, 5) by column indices of leading 1s in R.(85

)= 56 cases for columns J = {j1, . . . , j5} of leading 1s.

Ri ,ji = 1 (1 ≤ i ≤ 5), Ri ,k = 0 (k < ji ), Rk,ji = 0 (k 6= i).

If j > ji and j /∈ {j1, . . . , j5} then Ri ,j is an indeterminate.

J = {1, . . . , 5} (15 indets); J = {4, . . . , 8} (no indets).

I’ll discuss in detail J = {1, 4, 5, 6, 8}, since it is a case for which

the results are non-trivial, and

the computations fit on the screen.

Partition Gr(8, 5) by column indices of leading 1s in R.

For J = {1, 4, 5, 6, 8} the relation matrix is

1 x1 x2 . . . x3 .. . . 1 . . x4 .. . . . 1 . x5 .. . . . . 1 x6 .. . . . . . . 1

The rows of R represent these relations in degree 3:

(a1•a2)•a3 + x1(a1•a2)◦a3 + x2(a1◦a2)•a3 + x3a1◦(a2•a3) ≡ 0,

(a1◦a2)◦a3 + x4a1◦(a2•a3) ≡ 0,

a1•(a2•a3) + x5a1◦(a2•a3) ≡ 0,

a1•(a2◦a3) + x6a1◦(a2•a3) ≡ 0,

a1◦(a2◦a3) ≡ 0.

We want to generate all their consequences in degree 5.

1 x1 x2 . . . x3 .. . . 1 . . x4 .. . . . 1 . x5 .. . . . . 1 x6 .. . . . . . . 1

(a1◦a2)◦a3 + x4a1◦(a2•a3) ≡ 0,

a1•(a2•a3) + x5a1◦(a2•a3) ≡ 0,

a1•(a2◦a3) + x6a1◦(a2•a3) ≡ 0,

a1◦(a2◦a3) ≡ 0.

1 x1 x2 . . . x3 .. . . 1 . . x4 .. . . . 1 . x5 .. . . . . 1 x6 .. . . . . . . 1

(a1◦a2)◦a3 + x4a1◦(a2•a3) ≡ 0,

a1•(a2•a3) + x5a1◦(a2•a3) ≡ 0,

a1•(a2◦a3) + x6a1◦(a2•a3) ≡ 0,

a1◦(a2◦a3) ≡ 0.

There are 40 monomials in degree 4:

5 ways to place parentheses (Catalan number),

2 choices for each of 3 operations,

5 · 23 = 40.

Any relation f (a, b, c) in degree 3 has 10 consequences in degree 4:

f (a∗d , b, c), f (a, b∗d , c), f (a, b, c∗d), f (a, b, c)∗d , d∗f (a, b, c),

where ∗ ∈ {•, ◦}.The five relations for J = {1, 4, 5, 6, 8} produce 49 distinctconsequences (one repetition).

The consequences in degree 4 of the relations R in degree 3 arerepresented by a matrix of size 49× 40.

Its partial Smith form has an identity block of size 35 and a lowerright block of size 14× 5.

5 · 23 = 40.

where ∗ ∈ {•, ◦}.

The five relations for J = {1, 4, 5, 6, 8} produce 49 distinctconsequences (one repetition).

5 · 23 = 40.

The lower right block contains 6 zero rows and 1 zero column.

After deleting the zero rows and column we obtain (the transposeof) this 4× 8 matrix B:

0 0 0 0 x1 − x2 0 0 00 0 0 x5 − x1 0 −x6 −x3 x4−x6 −x1x2x6 x4 0 x21x2x4 0 0 0

0 0 −x4 0 −x1x22x4 0 −x1x2x6 −x6

We want nullity 4 for the original 49× 40 matrix, hence rank 36,and since the identity block has size 35, we want rank 1 for B:

DI1(B) 6= {0}, DI2(B) = {0}.

0 0 −x4 0 −x1x22x4 0 −x1x2x6 −x6

DI1(B) 6= {0}, DI2(B) = {0}.

0 0 −x4 0 −x1x22x4 0 −x1x2x6 −x6

DI1(B) 6= {0}, DI2(B) = {0}.

DI1(B) is the ideal generated by the entries of the matrix; withx1 ≺ x2 ≺ x3 ≺ x4 ≺ x5 ≺ x6, its degrevlex Grobner basis is

DI1(B) = ( x2 − x1, x3, x4, x5 − x1, x6 ) =√

DI1(B).

Thus B = O if and only if x1 = x2 = x5 and x3 = x4 = x6 = 0.

For D2(B) we need to compute all 2× 2 minors; ignoring 0 andmaking the rest monic, we obtain these 29 polynomials, sortedaccording to the monomial order:

x3x2 − x3x1, x4x2 − x4x1, x5x2 − x5x1 − x2x1 + x21 ,x6x2 − x6x1, x4x3, x6x3, x24 , x5x4 − x4x1, x6x4,x6x5 − x6x1, x26 , x6x

22x1 − x6x2x

21 , x6x3x2x1, x6x4x2x1,

x6x4x2x1 + x6x3, x6x5x2x1 − x6x2x21 , x26x2x1, x4x3x2x

x24x2x21 , x5x4x2x

21 − x4x2x

31 , x6x4x2x

21 , x4x3x

x24x22x1, x24x

22x1 − x24x2x

21 , x5x4x

22x1 − x4x

21 , x6x4x

x26x22x

21 , x6x4x

31 , x6x4x

DI1(B) = ( x2 − x1, x3, x4, x5 − x1, x6 ) =√

DI1(B).

22x1 − x6x2x

21 , x6x3x2x1, x6x4x2x1,

x24x2x21 , x5x4x2x

21 − x4x2x

31 , x6x4x2x

21 , x4x3x

x24x22x1, x24x

22x1 − x24x2x

21 , x5x4x

22x1 − x4x

21 , x6x4x

x26x22x

21 , x6x4x

31 , x6x4x

DI1(B) = ( x2 − x1, x3, x4, x5 − x1, x6 ) =√

DI1(B).

22x1 − x6x2x

21 , x6x3x2x1, x6x4x2x1,

x24x2x21 , x5x4x2x

21 − x4x2x

31 , x6x4x2x

21 , x4x3x

x24x22x1, x24x

22x1 − x24x2x

21 , x5x4x

22x1 − x4x

21 , x6x4x

x26x22x

21 , x6x4x

31 , x6x4x

DI1(B) = ( x2 − x1, x3, x4, x5 − x1, x6 ) =√

DI1(B).

22x1 − x6x2x

21 , x6x3x2x1, x6x4x2x1,

x24x2x21 , x5x4x2x

21 − x4x2x

31 , x6x4x2x

21 , x4x3x

x24x22x1, x24x

22x1 − x24x2x

21 , x5x4x

22x1 − x4x

21 , x6x4x

x26x22x

21 , x6x4x

31 , x6x4x

The degrevlex Grobner basis (unfactored) for DI2(B) is

x3x2 − x3x1, x4x2 − x4x1, x5x2 − x5x1 − x2x1 + x21 ,x6x2 − x6x1, x4x3, x6x3, x24 , x5x4 − x4x1, x6x4,x6x5 − x6x1, x26 .

DI2(B) is not a radical ideal; the Grobner basis for√DI2(B) is

x4, x6, (x2 − x1)x3, (x5 − x1)(x2 − x1).

This gives two families of solutions:

x1 = x5, x2 = free, x3 = 0, x4 = 0, x5 = free, x6 = 0

x1 = x2, x2 = free, x3 = free, x4 = 0, x5 = free, x6 = 0

For these values of the parameters we have rank(B) ≤ 1.

x4, x6, (x2 − x1)x3, (x5 − x1)(x2 − x1).

To get rank(B) = 1 we must exclude the solutions in the zero setof DI0(B), namely x1 = x2 = x5 and x3 = x4 = x6 = 0.

We have only checked degree 4; it remains to check degrees n ≥ 5.

This is only one of the easier cases out of a total of 56 cases forfive relations in degree 3 on two binary operations.

So the original question of the uniqueness of the diassociativerelations is still open!

Matrices and Submodules of Free Modules

Let A be an m × n matrix over P = F[x1, . . . , xk ], k ≥ 2.

The row vectors of A belong to Pn, the free P-module of rank n.

Row module of A = submodule of Pn generated by rows of A.

Hence the row module of A is a submodule of a free P-module.

Submodules of a free P-module of rank 1 are simply ideals in P.

The very useful theory of Grobner bases can be regarded as atheory of submodules of free P-modules of rank 1.

This was extended by Moller & Mora in 1986 to Grobner bases forsubmodules of free P-modules of rank n (J. Algebra 100, 138–178).

Computing the Grobner basis for the row module of the matrix A isessentially the same as computing a row canonical form for A (withrespect to a given monomial order and order of the columns).

Matrix form of the algorithm to compute the Grobner basis of asubmodule of a free module over a polynomial ring:

Input: An m× n matrix A with entries in P = F[x1, . . . , xk ], k ≥ 2,and a monomial order (extended to polynomials) f ≺ g on P.

Output: The Grobner basis for the row module of A, that is therow canonical form of A, with respect to the given order of thecolumns and the given monomial order on P.

Set i ← 1, j ← 1.

While i ≤ m and j ≤ n do:

If all entries at and below pivot (i , j) are 0 then set j ← j + 1.

Otherwise:

1 Repeat until convergence: Use row operations toswap the smallest nonzero entry into the pivot andreduce the other entries modulo the pivot.

2 Sort the entries at and below the pivot in increasing order,with 0 being the greatest.

3 For k = 1, . . . ,m − j repeat steps [1] and [2] for the entries atand below position (i + k, j) to self-reduce the column.

4 For every pair of indices k, k ′ such that i ≤ k 6= k ′ ≤ m andthe entries in positions (i , k) and (i , k ′) produce anS-polynomial with a nonzero reduced form modulo the entriesin rows i through m, do the following:

• Set m← m + 1; add a new zero row at the bottom.

• Use row operations to construct the S-polynomial inposition (m+1, j).

• Compute its nonzero reduced form modulo the entries inrows i through m.

5 Repeat steps [1]–[4] until the entries at and below the pivotform a reduced Grobner basis for the ideal they generate.

6 Delete any zero rows and modify m accordingly.

7 Use the Grobner basis at and below the pivot to reduce theentries above the pivot to their normal forms.

8 Set i ← i + 1, j ← j + 1.

Example 4

Consider the 10× 14 matrix A displayed in two parts below:

0 0 0 −ba2−a2 0 ba−a3 0

b2a2+ba2 b3a+b2a b2a2−2ba4+a6 b2a4+b2a2 b3a3−ba3 b3a−b2a3+ba5+ba3 0

0 0 0 0 −ba2−a2 0 0

0 0 −ba2+a4 0 0 −b2a+ba3 00 0 0 0 0 0 0

0 0 0 0 0 0 ba−a3

0 0 b2a2−2ba4+ba2+a6−a4 b2a4+ba4 b3a3+b2a3 b3a−b2a3+b2a+ba5 −b2a+ba3

0 0 ba−a3 ba3+a3 b2a2+ba2 0 0

0 0 ba−a3 −b2a−ba −b3−b2 −ba2−a2 0

−b2a−ba −b3−b2 ba3−a5 −b2a3−b2a −b3a2+ba2 −ba4−ba2 0

0 −b2a−ba 0 0 0 b3a+b2a b4+b3

0 −b4a+b3a3 b4a2−b2a2 −b4a−b3a b2a2−ba4+ba2−a4 b5a−b3a b6−b4

0 0 −b2a−ba 0 ba−a3 −b3−b2 0

0 b3a−b2a3 −b3a2+ba2 b3a+b2a −ba2−a2 −b4a+b2a −b5+b3

0 ba−a3 −ba2−a2 0 0 −b2a−ba −b3−b2

−ba2−a2 0 0 −b2a−ba 0 0 −b3−b2

b2a2+ba2 −b4a+b3a3−b3a+b2a3 b4a2+b3a2 −b4a−b3a b2a2−ba4 b5a+b4a b6+b5

0 b2a2+ba2 b3a+b2a −b3−b2 0 0 00 0 0 0 0 0 0

0 0 0 0 ba3+a3 0 0

Example 4

0 0 0 −ba2−a2 0 ba−a3 0

0 0 0 0 −ba2−a2 0 0

0 0 −ba2+a4 0 0 −b2a+ba3 00 0 0 0 0 0 0

0 0 0 0 0 0 ba−a3

0 0 ba−a3 ba3+a3 b2a2+ba2 0 0

0 0 ba−a3 −b2a−ba −b3−b2 −ba2−a2 0

0 −b2a−ba 0 0 0 b3a+b2a b4+b3

0 0 −b2a−ba 0 ba−a3 −b3−b2 0

0 ba−a3 −ba2−a2 0 0 −b2a−ba −b3−b2

−ba2−a2 0 0 −b2a−ba 0 0 −b3−b2

0 b2a2+ba2 b3a+b2a −b3−b2 0 0 00 0 0 0 0 0 0

0 0 0 0 ba3+a3 0 0

Example 4

0 0 0 −ba2−a2 0 ba−a3 0

0 0 0 0 −ba2−a2 0 0

0 0 −ba2+a4 0 0 −b2a+ba3 00 0 0 0 0 0 0

0 0 0 0 0 0 ba−a3

0 0 ba−a3 ba3+a3 b2a2+ba2 0 0

0 0 ba−a3 −b2a−ba −b3−b2 −ba2−a2 0

0 −b2a−ba 0 0 0 b3a+b2a b4+b3

0 0 −b2a−ba 0 ba−a3 −b3−b2 0

0 ba−a3 −ba2−a2 0 0 −b2a−ba −b3−b2

−ba2−a2 0 0 −b2a−ba 0 0 −b3−b2

0 b2a2+ba2 b3a+b2a −b3−b2 0 0 00 0 0 0 0 0 0

0 0 0 0 ba3+a3 0 0

Column 1. The first column has two nonzero entries whichgenerate the principal ideal (ab2 + ab).

We interchange rows 1 and 10, multiply row 1 by −1, and thensubtract a times row 1 from row 2.Column 1 is now zero except for the ideal generator in row 1.

Column 2. At this point column 2 has b3 + b2 in row 1 and zerosin the other rows, so it is already reduced.

Column 3. The entries in column 3 in row 2 and below generatethe principal ideal (ba− a3):

ba− a3, −ba2 + a4, b2a2−ba4, b2a2− 2ba4 +ba2 + a6− a4.

Conveniently, the generator appears in row 8, so we swap it up torow 2 and use row operations to eliminate the entries below it.The entry in row 1 is −ba3 + a5, which is −a2 times the generator;we use one more row operation to make this entry zero too.

Column 1. The first column has two nonzero entries whichgenerate the principal ideal (ab2 + ab).We interchange rows 1 and 10, multiply row 1 by −1, and thensubtract a times row 1 from row 2.

Column 1 is now zero except for the ideal generator in row 1.

Column 1. The first column has two nonzero entries whichgenerate the principal ideal (ab2 + ab).We interchange rows 1 and 10, multiply row 1 by −1, and thensubtract a times row 1 from row 2.Column 1 is now zero except for the ideal generator in row 1.

Conveniently, the generator appears in row 8, so we swap it up torow 2 and use row operations to eliminate the entries below it.

The entry in row 1 is −ba3 + a5, which is −a2 times the generator;we use one more row operation to make this entry zero too.

Column 4. The entries in column 4 in row 3 and below generateour first non-principal ideal:

−ba2 − a2, ba4 + a4, ba6 − ba4 + a6 − a4,−b2a− ba3 − ba− a3, −b2a4 − ba4.

The pure lex (a ≺ b) Grobner basis for this ideal is

ba2 + a2, b2a + ba.

The first of these polynomials already appears in the column sowe swap it up to row 3 and use row operations to replace each ofthe lower entries by their remainders modulo this entry.This makes all the lower entries zero except for −b2a− ba (row 9);we swap it up to row 4 and change its sign.We now have the Grobner basis in rows 3 and 4; we use it toreduce the entries in rows 1 and 2.

ba2 + a2, b2a + ba.

The first of these polynomials already appears in the column sowe swap it up to row 3 and use row operations to replace each ofthe lower entries by their remainders modulo this entry.

This makes all the lower entries zero except for −b2a− ba (row 9);we swap it up to row 4 and change its sign.We now have the Grobner basis in rows 3 and 4; we use it toreduce the entries in rows 1 and 2.

ba2 + a2, b2a + ba.

The first of these polynomials already appears in the column sowe swap it up to row 3 and use row operations to replace each ofthe lower entries by their remainders modulo this entry.This makes all the lower entries zero except for −b2a− ba (row 9);we swap it up to row 4 and change its sign.

We now have the Grobner basis in rows 3 and 4; we use it toreduce the entries in rows 1 and 2.

ba2 + a2, b2a + ba.

Column 5. The entries in column 5, in row 5 and below, generatethe principal ideal (ba2 + a2). The negative of the generator is theentry in row 10, so we swap it up to row 5, change its sign, anduse row operations to eliminate the entries below it and reduce theentries above it.

The calculations get significantly more complicated at this point,so we record the state of the reduced part of the matrix after thereduction of column 5. The upper left 5× 5 block is as follows,and the 5× 5 block below it is the zero matrix:

b2a + ba b3 + b2 0 −ba + a3 −b3 − b2

0 0 ba− a3 0 00 0 0 ba2 + a2 00 0 0 b2a + ba b3 + b2

0 0 0 0 ba2 + a2

b2a + ba b3 + b2 0 −ba + a3 −b3 − b2

0 0 ba− a3 0 00 0 0 ba2 + a2 00 0 0 b2a + ba b3 + b2

0 0 0 0 ba2 + a2

b2a + ba b3 + b2 0 −ba + a3 −b3 − b2

0 0 ba− a3 0 00 0 0 ba2 + a2 00 0 0 b2a + ba b3 + b2

0 0 0 0 ba2 + a2

Column 6. The nonzero entries at or below the current pivot (6,6)are as follows, appearing once each in rows 9, 8, 7 respectively :

f = −b2a + 2ba3 − a5,

g = b3a− 2b2a3 + ba5,

h = b3a− b2a3 + b2a + 2ba5 − ba3 − a7 + a5.

Clearly g = −bf so we eliminate g by a row operation with f .

Replacing f by −f , we get these generators of the column ideal:

f = b2a−2ba3+a5, g = b3a−b2a3+b2a+2ba5−ba3−a7+a5.

The normal form of g modulo f is 3ba5 + ba3 − 2a7, so the newgenerators are (renaming again):

f = 3ba5 + ba3 − 2a7, g = b2a− 2ba3 + a5.

f = −b2a + 2ba3 − a5,

g = b3a− 2b2a3 + ba5,

h = b3a− b2a3 + b2a + 2ba5 − ba3 − a7 + a5.

Clearly g = −bf so we eliminate g by a row operation with f .Replacing f by −f , we get these generators of the column ideal:

f = 3ba5 + ba3 − 2a7, g = b2a− 2ba3 + a5.

f = −b2a + 2ba3 − a5,

g = b3a− 2b2a3 + ba5,

h = b3a− b2a3 + b2a + 2ba5 − ba3 − a7 + a5.

Clearly g = −bf so we eliminate g by a row operation with f .Replacing f by −f , we get these generators of the column ideal:

f = 3ba5 + ba3 − 2a7, g = b2a− 2ba3 + a5.

We now have a self-reduced set, so we consider S-polynomials.

There is one, denoted s, corresponding to the overlap ab.

We also give its reduced form s ′ modulo f and g :

s = b2a3 + 4ba7 − 3a9, s ′ = 2ba3 + 3a9 + 5a7.

Computing the reduced forms of f and g modulo s ′ gives thepolynomials f ′ and g ′: