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### 8 Derivations

A derivation d of a ring R is a map R -> R that obeys the product rule

d(rs) = d(r)s + rd(s)

for elements r, s in the field, and which is also distributive over addition in the ring

d(r+s) = d(r) + d(s)

#### 8.1 The `HAPDerivation` datatype

The `HAPDerivation` datatype represents a derivation of a polynomial ring R. Given the product and addition rules above, a derivation can be completely specified in terms of the images of the ring indeterminates. Consequently, a `HAPDerivation` is specified by providing:

• a polynomial ring R

• a list of images, under the derivation, of each of the ring's indeterminates.

Optionally, you can also provide

• a set of relations I (which is assumed to be empty if none is provided).

The support for derivations over quotient rings using `HAPDerivation` is limited. The ring is always assumed to simply be the polynomial ring R, but when calculating the kernel and homology (see `KernelOfDerivation` (8.5-2) and `HomologyOfDerivation` (8.5-3)), any relations in I are also included in the result's relations, which is the equivalent of calculating the kernel of homology of the derivation R/I -> R/I.

#### 8.2 Computing the kernel and homology of a derivation

Computing the kernel of a derivation is difficult in the general case, but for polynomial rings over F_p we have a novel solution which allows derivations to be considered as module homomorphisms.

Consider a ring R over GF(2). For any r in R,

d(r^2) = d(rr) = d(r)r + rd(r) = 2d(r) = 0

In general, for a ring R = F_p[x_1, ..., x_n], the subring S = F_p[x_1^p, ..., x_n^p] is in the kernel of d. For this choice of S, R can always be written as a free S-module with 2^n generators.

Consider now taking the derivation of sr where s in S and r in R,

d(sr) = sd(r) + d(s)r = sd(r)

since s is in the kernel of d. In other words, when R is written as an S-module, the derivation d behaves exactly as as an S-module homomorphism.

The Singular computer algebra system can compute the kernels of module homomorphisms over polynomial rings, and the singular package provides an interface between Singular and GAP. The function `KernelOfDerivation` (8.5-2) constructs S, expresses d as an S-module homomorphism (as well as converting any ideal), and asks Singular for the kernel. Singular returns the result as a set of module generators, which are then reduced to a set of ring generators and a presentation found. This makes further use of Singular, and the `HAPRingHomomorphism` datatype (see Chapter 7).

The homology of a derivation is defined to be

Hom(d) = \ker (d) / im (d)

Given the kernel of a derivation and the image (expressed as the images of the module generators), the homology can be readily computed, as is done by `HomologyOfDerivation` (8.5-3). The natural presentation obtained by concatenating the kernel relations and the image typically contains redundancies and the `HAPRingReductionHomomorphism` (7.2-4) is used to generate a nicer presentation involving fewer indeterminates and relations.

#### 8.3 Construction function

##### 8.3-1 HAPDerivation construction functions
 `> HAPDerivation`( R[, I], images ) ( operation )
 `> HAPDerivationNC`( R, I, images ) ( operation )

Returns: `HAPDerivation`

Construct a `HAPDerivation` object representing the derivation d where R is a polynomial ring and images is the list of the images of the ring indeterminates under the derivation, d(x_1), d(x_2), ..., d(x_n). An optional set of relations I can also be provided, which are passed to `KernelOfDerivation` (8.5-2) when calculating the kernel or homology of this derivation. The function `HAPDerivation` checks that the arguments are compatible, while the `NC` method performs no checks.

#### 8.4 Data access function

##### 8.4-1 DerivationRing
 `> DerivationRing`( d ) ( attribute )

Returns: Polynomial ring

Returns the ring over which the derivation d is defined.

##### 8.4-2 DerivationImages
 `> DerivationImages`( d ) ( attribute )

Returns: List of polynomials

A derivation d over a (quotient) polynomial ring is defined by a set of images. This function returns this list of images. The ith element of the list is the image of indeterminate number i in that ring family. (Note that indeterminate number i is not necessarily the ith indeterminate in that particular ring. See Reference: Indeterminates for more details.)

##### 8.4-3 DerivationRelations
 `> DerivationRelations`( d ) ( attribute )

Returns: List of polynomials

Returns the relations of the quotient ring over which the derivation d is defined.

##### 8.4-4 Example: Constructing and accessing data of a `HAPDerivation`

We demonstrate creating a `HAPDerivation` object and reading back its data by creating a derivation d of the polynomial ring over the integers with two indeterminates, Z[x_1, x_2, x_3], which has the mapping d(x_1) = d(x_2) = 1, d(x_3) = 0.

 ```gap> ring := PolynomialRing(Integers, 3);; gap> d := HAPDerivation(ring, [One(ring), One(ring), Zero(ring)]); gap> gap> DerivationRing(d); Integers[x_1,x_2,x_3] gap> DerivationImages(d); [ 1, 1, 0 ] gap> DerivationRelations(d); [ ] ```

#### 8.5 Image, kernel and homology functions

##### 8.5-1 ImageOfDerivation
 `> ImageOfDerivation`( d, poly ) ( operation )

Returns: Polynomial

Returns the image of the polynomial poly under the derivation d. (poly must be a polynomial in the derivation's ring.)

##### 8.5-2 KernelOfDerivation
 `> KernelOfDerivation`( d[, avoid] ) ( operation )

Returns: List

Returns a ring presentation S/J for the kernel of the derivation d, where S is a polynomial ring and J are a set of relations. This operation returns a list with the following elements:

1. the new polynomial ring S

2. a basis for the ideal, as a set of relations J

3. the ring isomorphism between the kernel (i.e. a subring of the derivation's ring) and the new ring. This is given using the `HAPRingHomomorphism` type, for details of which see 7

An optional parameter, avoid, can be provided which lists indeterminates to avoid when creating the the new polynomial ring.

This function is only available if the package singular is available.

##### 8.5-3 HomologyOfDerivation
 `> HomologyOfDerivation`( d[, avoid] ) ( operation )

Returns: List

Returns a ring presentation S/J for the homology of the derivation d, where S is a polynomial ring and J are a set of relations. Returns a polynomial ring presentation for the homology ker(d)/im(d) of the derivation d. This operation returns a list with the following elements:

1. the new polynomial ring S

2. a basis for the ideal, as a set of relations J

3. the ring isomorphism between the kernel (i.e. a subring of the derivation's ring) and the new ring. This is given using the `HAPRingHomomorphism` type, for details of which see 7

An optional parameter, avoid, can be provided which lists indeterminates to avoid when creating the the new polynomial ring.

##### 8.5-4 Example: Homology of a `HAPDerivation`

In this example, we wish to calculate the homology of a derivation d on the ring F_2[x,y,z]/(yx^2 + xy^2) where d(x) = d(y) = 0 and d(z) = yx^2 + xy^2. To demonstrate some of the other functions, we also calculate the kernel of this derivation and the image of the element xz.

 ```gap> ring := PolynomialRing(GF(2), 3);; gap> x := ring.1;; y := ring.2;; z := ring.3;; gap> d := HAPDerivation(ring, [x^2 + x*y + y^2], > [Zero(ring), Zero(ring), x^2*y + x*y^2]); gap> # gap> ImageOfDerivation(d, x*z); x_1^3*x_2+x_1^2*x_2^2 gap> KernelOfDerivation(d); [ GF(2)[x_7,x_8,x_9], [ x_7^2+x_7*x_8+x_8^2 ], ] gap> HomologyOfDerivation(d); [ GF(2)[x_4,x_5,x_6], [ x_4^2+x_4*x_5+x_5^2, x_5^3 ], ] ```

The result we were after, the homology of d, can be read from the final result as follows. The first two entries in the list give the homology group as a quotient ring: the quotient of the polynomial ring F_2[x_6,x_7,x_8] with the two relations (x_7^2+x_7x_8+x_8^2, x_8^3). The last entry in the list gives the new indeterminates in terms of the original indeterminates, and from this we know that it would be friendlier to write the result as

F_2[x,y,e^2] / (x^2+xy+y^2, y^3)

The kernel of the derivation can read from the result in a similar manner.

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