Copyright (c) 2005 John Abbott
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.2;
with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
A copy of the licence is included in the file COPYING in this directory.
User documentation for the files ring.H and ring.C
The file ring.H introduces several classes used for representing rings
and their elements. A normal user of the CoCoA library will use mostly
the classes [ring] and [RingElem]. An object of type [ring] represents
a mathematical ring with unity, and objects of type [RingElem] represent
values from some ring. While the CoCoA library was conceived primarily
for computing with commutative rings, the possibility of having certain
non-commutative rings exists.
A variable of type [ring] is effectively constant: its value may not be
changed, and it cannot be assigned to. In contrast, a variable of type
[RingElem] may change its value but only within the ring to which it is
associated: this ring is specified when the variable is created. Here
are a couple of code snippets to give an idea:
VALID CODE INVALID CODE
ring R = RingZ(); ring R;
RingElem a(R), b(R);
a = 2; RingElem a = 2;
b = a+7; RingElem(2);
if (a-b != -7) cout << "MISTAKE!\n";
There are fuller example programs in the "examples/" directory.
Operations on Rings
The primary use for a variable of type [ring] is simply to specify
the ring to which RingElem variables are associated. However, for some
computations the best algorithm to use may depend on certain properties
of the ring. Let R denote a variable of type [ring]: here are the
properties you can ask about:
characteristic(R) -- returns the characteristic of R (as a ZZ)
IsCommutative(R) -- a boolean, true iff R is commutative
IsIntegralDomain(R) -- a boolean, true iff R has no zero divisors
IsGCDDomain(R) -- a boolean, true iff R is a GCD domain
IsOrderedDomain(R) -- a boolean, true iff R is arithmetically ordered
IsField(R) -- a boolean, true iff R is a field
symbols(R) -- a std::vector of the symbols in R
(e.g. Q[x,y,z] contains the symbols x,y, and z)
NOTE: a pragmatic approach is taken: e.g. [IsGCDDomain] is true only if
GCDs can actually be computed using the CoCoA library. [IsQuotientRing]
is true only if R is internally implemented as a quotient ring
(e.g. Z/0Z is; Z is not). In general the function "IsXYZ" should be
read as "Is internally implemented as XYZ".
These query functions tell you how the ring is implemented:
IsZ(R) -- a boolean, true iff R is actually a ring of integers
IsFractionField(R) -- a boolean, true iff R is actually a field of fractions
IsPolyRing(R) -- a boolean, true iff R is actually a polynomial ring
IsQuotientRing(R) -- a boolean, true iff R is actually a quotient ring
NOTE: Warning: for example if R = Z[x]/ideal(x) then
R is obviously isomorphic to the ring of integers but
IsZ(R) gives FALSE, and
IsQuotientRing(R) gives TRUE.
You can get the zero and one of a ring directly using the following:
zero(R) -- the zero element of R
one(R) -- the one element of R
Operations on RingElems
A variable of type [RingElem] must be associated to a ring upon its
creation. Thereafter the ring to which it is associated cannot be
changed during the lifetime of the value, but the value inside the ring
can be changed. [RingElems] are designed to be easy and safe to use;
this incurs a certain run-time overhead, so a faster alternative is
offered (see below in section "Fast and Ugly Code").
See also some example programs in the directory "examples/" for how to
Constructors of [RingElem]s:
assume R is a [ring],
n a machine integer, and
N a (big) integer of type [ZZ]
RingElem r1(R); -- an element of R, initially zero
RingElem r2(R, n); -- an element of R, initially the image of n
RingElem r3(R, N); -- an element of R, initially the image of N
RingElem r4(r1); -- a copy of r1, element of the same ring
RingElem r4 = r1; -- (alternative syntax, discouraged)
zero(R) -- the zero element of R
one(R) -- the one element of R
let r1 be a (possibly const) [RingElem], and
let N be a variable of type ZZ
owner(r1) -- the ring to which r1 is associated
IsZero(r1) -- true iff r1 is zero
IsOne(r1) -- true iff r1 is one
IsMinusOne(r1) -- true iff -r1 is one
IsInteger(N, r1) -- true iff r1 is the image of an integer
(if true, a preimage is placed in N)
IsRational(N, D, r1) -- true iff r1 is the image of a rational
(if true, N/D becomes a preimage)
IsUnit(r1) -- true iff r1 has a multiplicative inverse
IsDivisible(r1, r2) -- true iff r1 is divisible by r2
raw(r1) -- (see below, section "Fast and Ugly Code")
Arithmetic: let r be a non-const [RingElem], and r1, r2 be potentially
const [RingElems]. Assume they are all associated to the same ring.
Then the operations available are: (meanings are obvious)
cout << r1 -- output value of r1 (decimal only, see notes)
r1 == r2 -- equality test
r1 != r2 -- not-equal test
r = r1 -- assignment
-r1 -- negation (unary minus)
r1 + r2 -- sum
r1 - r2 -- difference
r1 * r2 -- product
r1 / r2 -- quotient, division must be exact (see IsDivisible)
r += r1 -- equiv. r = r + r1
r -= r1 -- equiv. r = r - r1
r *= r1 -- equiv. r = r * r1
r /= r1 -- equiv. r = r / r1
gcd(r1, r2) -- an associate of the gcd (if ring is gcd domain)
power(r1, n) -- n-th power of r1; n may be [long] or [ZZ]
Ordering: if the ring is ordered then these functions may also be used.
sign(r1) -- value is -1, 0 or +1 according as r1 is
negative, zero or positive
[here r1 must belong to an ordered ring]
abs(r1) -- absolute value of r1
cmp(r1, r2) -- returns a value <0, =0, >0 according as
r1-r2 is <0, =0, >0.
r1 < r2 -- standard inequalities
r1 > r2 -- ...
r1 <= r2 -- ...
r1 >= r2 -- ...
Notes on operations
Operations on elements of different rings will cause a run-time error.
In all functions involving two [RingElems] either r1 or r2 may be replaced
by a machine integer (the type is actually [long]), or by a big integer
(element of the class ZZ). The integer value is automatically mapped
into the ring owning the [RingElem] in the same expression.
The exponent n in the power function may be zero or negative, but a
run-time error will be signalled if one attempts to compute a negative
power of a non-invertible element or if one attempts to raise zero to
the power zero. You cannot use [^] to compute powers -- see Bugs section.
An attempt to perform an inexact division or to compute a GCD not in a
GCD domain will produce a run-time error.
The printing of ring elements is always in decimal regardless of the
[ostream] settings (this is supposed to be a feature rather than a bug).
At this point, if you are new to CoCoALib, you should probably look
at some of the example programs in the "examples/" directory.
Writing functions with [RingElems] as arguments
One would normally expect to use the type [const RingElem&] for
read-only arguments which are [RingElems], and [RingElem&] for
read-write arguments. Unfortunately, doing so would lead
to problems with the CoCoA library. INSTEAD you should use the type
[ConstRefRingElem] for read-only arguments, and the type [RefRingElem]
for read-write arguments.
If you are curious to know why this non-standard quirk has to be used,
read on. Internally, ring element values are really smart pointers to
the true value. Now the "const" keyword in C++ when applied to a
pointer makes the pointer const while the pointed-to value remains
alterable -- this is not the behaviour we want for [const RingElem&].
To get the desired behaviour we have to use another type: the type
we haved called [ConstRefRingElem].
You might wonder why "Ref" appears in the names [RefRingElem] and
[ConstRefRingElem]. It indicates that you are working with a reference
to a value which is owned by another object (e.g. a variable of type
[RingElem], or maybe a matrix).
ADVANCED USE OF RINGS
The rest of this section is for more advanced use of rings and RingElems
(e.g. by CoCoA library contributors). If you are new to CoCoA, you need
not read beyond here.
Writing C++ classes for new types of ring
A convention of the CoCoA library is that the class RingBase is to be
used as an abstract base class for all rings. You are strongly urged to
familiarize yourself with the maintainer documentation if you want to
understand how and why rings are implemented as they are in the CoCoA
Fast and Ugly Code
WE DO NOT RECOMMEND that you use what is described in this section.
If you are curious to know a bit more how rings are implemented, you
might find this section informative.
RingElems are designed to be easy and pleasant to use, but this
convenience has a price: a run-time performance penalty (and a memory
space penalty too). Both penalities may be avoided by using "raw
values" but at a considerable loss of programming convenience and
safety. You should consider using raw values only if you are desperate
for speed; even so, performance gains may be only marginal except
perhaps for operations on elements of a simple ring (e.g. a small finite
A RingElem object contains within itself an indication of the owning
ring, and a "raw value" which is a pointer to where the real
representation of the ring element value lies. These raw values may be
accessed via the "raw" function. They may be combined arithmetically by
calling member functions of the owning ring. For instance, if x,y,z are
all RingElem objects all BELONGING TO EXACTLY THE SAME RING then we can
x = y+z;
slightly faster by calling
owner(x)->my.Add(raw(x), raw(y), raw(z));
It should now be clear that the syntax involved is cumbersome and
somewhat obscure. For the future maintainability of the code the
simpler "x = y+z;" has many advantages. Furthermore, should x,y,z
somehow happen not all to lie in the same ring then "x = y+z;" will act
in a reasonable way, whereas the supposedly faster call will likely lead
to many hours of debugging grief. The member functions for arithmetic
(e.g. myAdd) DO NOT PERFORM sanity checks on their arguments:
e.g. attempting to divide by zero could well crash the program.
If you use a "debugging version" of the CoCoA Library then some member
functions do use assertions to check their arguments. This is useful
during development, but must not be relied upon since the checks are
absent from the "non-debugging version" of the CoCoA Library. See the
file config.txt for more information.
This fast, ugly, unsafe way of programming is made available for those who
desperately need the speed. If you're not desperate, don't use it!
Fast, Ugly and Unsafe operations on raw values
Read the section "Fast and Ugly Code" before using any of these!
Let r be a non-const raw value, and r1, r2 potentially const raw values.
Assume they are all owned by the ring R. Then the functions available are:
R->myNew() -- "construct" a new element of R, value=0
R->myNew(n) -- "construct" a new element of R, value=n
R->myNew(N) -- "construct" a new element of R, value=N
R->myNew(r1) -- "construct" a new element of R, value=r1
R->myDelete(r) -- "destroy" r, element of R (frees resources)
R->mySwap(r, s) -- swaps the two values (s is non-const raw value)
R->myAssignZero(r) -- r = 0
R->myAssign(r, r1) -- r = r1
R->myAssign(r, n) -- r = n (n is a long)
R->myAssign(r, N) -- r = n (N is a ZZ)
R->myNegate(r, r1) -- r = -r1
R->myAdd(r, r1, r2) -- r = r1+r2
R->mySub(r, r1, r2) -- r = r1-r2
R->myMul(r, r1, r2) -- r = r1*r2
R->myDiv(r, r1, r2) -- r = r1/r2 (division must be exact)
R->myIsDivisible(r, r1, r2) -- r = r1/r2, and returns true iff division was exact
R->myIsUnit(r1) -- IsUnit(r1)
R->myGcd(r, r1, r2) -- r = gcd(r1, r2)
R->myPower(r, r1, n) -- r = power(r1, n) BUT n MUST be non-negative!!
R->myIsZero(r1) -- r1 == 0
R->myIsZeroAddMul(r, r1, r2) -- ((r += r1*r2) == 0)
R->myIsEqual(r1, r2) -- r1 == r2
R->myOutput(out, r1) -- out << r1
R->mySequentialPower(r, r1, n) -- normally it is better to use R->myPower(r, r1, n)
R->myBinaryPower(r, r1, n) -- normally it is better to use R->myPower(r, r1, n)
Maintainer documentation for the classes ring, RingBase and RingElem
(NB consider consulting also QuotientRing, FractionField and PolyRing)
The design underlying rings and their elements is more complex than I
would have liked, but it is not as complex as the source code may make
it appear. The guiding principles are that the implementation should be
flexible and easy/pleasant to use while offering a good degree of
safety; extreme speed of execution was not a goal (as it is usually
contrary to good flexibility) though an interface offering slightly
better run-time efficiency remains.
Regarding flexibility: in CoCoALib we want to handle polynomials whose
coefficients reside in (almost) any commutative ring. Furthermore, the
actual rings to be used will be decided at run-time, and cannot
restricted to a given finite set. We have chosen to use C++ inheritance
to achieve the implementation: the abstract class [RingBase] defines the
interface that every concrete ring class must offer.
Regarding ease of use: since C++ allows the common arithmetic operators
to be overloaded, it is essential that these work as expected for
elements of arbitrary rings -- with the caveat that "/" means exact
division, being the only reasonable interpretation. Due to problems of
ambiguity arithmetic between elements of different rings is forbidden:
e.g. let f in Q[x,y] and g in Z[y,x], where should f+g reside?
The classes in the file ring.H are closely interrelated, and there is no
obvious starting point for describing them -- you may find that you need
to read the following more than once to comprehend it. Here is a list
of the classes:
ring -- value represents a ring; it is a smart pointer
RingBase -- abstract class "defining what a ring is"
RingElem -- value represents an element of a ring
RefRingElem -- reference to a RingElem
ConstRefRingElem -- const-reference to a RingElem
RingElemConstRawPtr -- raw pointer to a "const" ring value
RingElemRawPtr -- raw pointer to a ring value
The class [RingBase] is of interest primarily to those wanting to
implement new types of ring (see relevant section below); otherwise you
probably don't need to know about it. Note that [RingBase] includes an
intrusive reference counter -- so every concrete ring instance will have
one. [RingBase] also includes a machine integer field containing a
unique numerical ID -- this is so that distinct copies of otherwise
identical rings can be distinguished when output (e.g. in OpenMath).
The class [ring] is used to represent mathematical rings (e.g. possible
values include Z, Q, or Q[x,y,z]). An object of type [ring] is just a
reference counting smart pointer to a concrete ring implementation
object -- so copying a ring is fairly cheap. In particular two rings
are considered equal if and only if they refer to the same identical
concrete ring implementation object. In other files you will find
classes derived from [ring] which represent special subclasses of rings,
for instance [PolyRing] is used to represent polynomial rings. The
intrusive reference count, which must be present in every concrete ring
implementation object, is defined as a data member of [RingBase].
The classes [RingElem], [RefRingElem] and [ConstRefRingElem] are related
by inheritance: they are very similar but differ in important ways. The
three classes are used for representing values in rings (e.g 1 as an
element Z, or 1 as an element of Q[x], etc). In each case an object of
that C++ type comprises two components: one is the identity of ring to
which the element belongs, and the other is the value in that ring (the
value is stored in a format that only the owning ring can comprehend).
All operations on ring elements are effected by member functions of the
ring to which the value belongs.
The difference between [ConstRefRingElem] and [RefRingElem] is quite
simple: you cannot change the value of a [ConstRefRingElem], for
instance you cannot assign to it, while you are free to change the value
of a [RefRingElem].
The difference between a [RefRingElem] and a [RingElem] is also quite
simple, but possibly harder to grasp. A variable of type [RingElem] is
the owner of the value that it represents: that value will be destroyed
when the variable passes out of scope. In contrast, a variable of type
[RefRingElem] is not the owner; it merely refers to value belonging to
some other structure (e.g. a [RingElem], or a matrix, or a polynomial).
So you can create a [RingElem] from nothing, whereas you must already
have a ring element to be able to create a [RefRingElem] which refers to
Why bother to distinguish between [RingElem] and [RefRingElem]? The
main reason was to allow matrices and iterators of polynomials to be
implemented cleanly and efficiently. Clearly a matrix should be the
owner of the values appearing as its entries, but we also want a way of
reading the matrix entries without having to copy them. Furthermore,
the matrix can use a compact representation: the ring to which its
elements belong is stored just once, and not once for each element.
The reason that [ConstRefRingElem] and [RefRingElem] are distinct
classes is that neither [const RefRingElem&] nor [const RingElem&]
achieves what one might reasonably expect. Since a [RingElem] is
effectively a pointer to the value represented, applying a C++ [const]
keyword merely makes the pointer const while leaving the pointed-to
value modifiable. Consider the following procedure
void CannotChange(const RefRingElem& x)
writable = writable+1;
The above procedure will add one to the value of its argument even
though it would seem that it should not be alter the value.
The inheritance structure between [ConstRefRingElem], [RefRingElem] and
[RingElem] implements the similarities and differences between these
classes while also allowing [ConstRefRingElem] and [RefRingElem] to be
used as types of parameters to functions and procedures. Given that
matrix entry accessors return a [ConstRefRingElem] it is important not
to use [RingElem&] or [const RingElem&] as the parameter type because
compilation would fail if a matrix entry were passed as parameter.
As already hinted above, the internal data layouts for objects of types
[RingElem], [RefRingElem] and [ConstRefRingElem] are identical -- this
is guaranteed by the C++ inheritance mechanism. The subfield indicating
the ring to which the value belongs is simply a [const ring], which is just a
reference counting smart pointer. The subfield indicating the value is
a raw pointer of type [void* const]; however, when the raw pointer value
is to be handled outside a ring element object then it is wrapped up as
a [RingElemRawPtr] or [RingElemConstRawPtr] -- these are simply wrapped
copies of the [void*]. Make a careful note of the exact type of the
data member [myValuePtr]: the pointer is constant while the pointed to
value is not constant. The constness of the pointer is ABSOLUTELY
CRUCIAL to the correct behaviour of [RefRingElem]. The fact that the
pointed-to value is not const may seem contradictory (for an object of
type [ConstRefRingElem]), but it allows slightly easier implementation
of the non-constant derived classes [RefRingElem] and [RingElem]; the
friend [raw] function puts in the necessary constness when it is called.
The classes [RingElemRawPtr] and [RingElemConstRawPtr] are used for two
reasons. One is that if a naked [void*] were used outside the ring
element objects then C++ would find the call [RingElem(R, 0)] ambiguous
because the constant  can be interpreted either as an integer
constant or as a null pointer: there are two constructors which match
the call equally well. The other reason is that it discourages
accidentally creating a ring element object from any old pointer; it
makes the programmer think -- plus I feel uneasy when there are naked
[void*] pointers around. Note that the type of the data member [myPtr]
is simply [void*] as opposed to [void const*] which one might reasonably
expect. I implemented it this way as it is simpler to add in the missing
constness in the member function [RingElemConstRawPtr::myRawPtr] than it
would be to cast it away in the [myRawPtr] function of [RingElemRawPtr].
In [ConstRefRingElem] why did I chose to make the data member [myValuePtr] of
type [void* const] rather than [RingElemRawPtr const]?
Further comments about implementation aspects of the above classes.
Recall that [ring] is essentially a smart pointer to a [RingBase]
object, i.e. concrete implementation of a ring. Access to the
implementation is given via [operator->]. If necessary, the pointer may
also be read using the member function [myRingPtr]: this is helpful for
defining functions such as [IsPolyRing] where access to the pointer is
required but [operator->] cannot be used.
The class [RingBase] declares a number of pure virtual functions for
computing with ring elements. Since these functions are pure they must
all be fully defined in any instantiable ring class (e.g. RingZImpl or
RingFpImpl). These member functions follow certain conventions:
* RETURN VALUES: most arithmetic functions return no value, instead the
result is placed in one of the arguments (normally the first argument
is the one in which the result is placed), but functions which return
particularly simple values (e.g. booleans or machine integers) do
indeed return the values by the usual function return mechanism.
* ARG TYPES: ring element values are passed as "raw pointers"
(i.e. a wrapped [void*] pointing to the actual value). A read-only
arg is of type [RingElemConstRawPtr], while a writable arg is of type
[RingElemRawPtr]. When there are writable args they normally appear
first. For brevity there are typedefs [ConstRawPtr] and [RawPtr] in
the scope of [RingBase] or any derived class.
* ARG CHECKS: sanity checks on the arguments are NOT CONDUCTED (e.g. the
division function assumes the divisor is non-zero). These member
functions are supposed to be fast rather than safe.
In a few cases there are non-pure virtual member functions in
[RingBase]. They exist either because there is a simple universal
definition or merely to avoid having to define inappropriate member
functions (e.g. gcd functions when the ring cannot be a gcd domain).
Here is a list of them:
IamGCDDomain() -- defaults to true
IamOrderedDomain -- defaults to false
myIsUnit(x) -- by default checks that 1 is divisible by x
myGcd(lhs, x, y) -- gives an error: either NotGcdDom or NYI
myGcdQuot(lhs, xquot, yquot, x, y) -- gives an error: either NotGcdDom or NYI
myExgcd(lhs, xcofac, ycofac, x, y) -- gives an error: either NotGcdDom or NYI
myIsPrintAtom(x) -- defaults to false
myIsPrintedWithMinus(x) -- gives a SERIOUS error
myIsMinusOne(x) -- defaults to myIsOne(-x); calculates -x
myIsZeroAddMul(lhs, y, z) -- computes lhs += y*z in the obvious way, and calls myIsZero
myCmp(x, y) -- gives NotOrdDom error
mySign(x) -- simply calls myCmp(x, 0), then returns -1,0,1 accordingly
There are three non-virtual member functions for calculating powers: one
uses the sequential method, the other two implement the repeated
squaring method (one is an entry point, the other an implementation
detail). These are non-virtual since they do not need to be redefined;
they are universal for all rings.
For the moment I shall assume that the intended meaning of the pure
virtual functions is obvious (given the comments in the source code).
Recall that arithmetic operations on objects of type [RingElem],
[RefRingElem] and [ConstRefRingElem] are converted into member function
calls of the corresponding owning ring. Here is the source code for
addition of ring elements -- it typifies the implementation of
operations on ring elements.
RingElem operator+(ConstRefRingElem x, ConstRefRingElem y)
const ring& Rx = owner(x);
const ring& Ry = owner(y);
if (Rx != Ry)
error(CoCoAError(ERR::MixedRings, "RingElem + RingElem"));
Rx->myAdd(raw(ans), raw(x), raw(y));
The arguments are of type [ConstRefRingElem] since they are read-only,
and the return type is [RingElem] since it is new self-owning value
(it does not refer to a value belinging to some other structure].
Inside the function we check that the rings of the arguments are
compatible, and report an error if they are not. Otherwise a temporary
local variable is created for the answer, and the actual computation is
effected via a member function call to the ring in which the values
lie. Note the use of the [raw] function for accessing the raw pointer
of a ring element. In summary, an operation on ring elements intended
for public use should fully check its arguments for compatibility and
correctness (e.g. to avoid division by zero); if all checks pass, the
result is computed by passing raw pointers to the appropriate member
functions of the ring involved -- this member function assumes that the
values handed to it are compatible and valid; if not, "undefined
behaviour" will result (i.e. a crash if you are lucky).
Most of the member functions of a ring are for manipulating raw values
from that same ring, a few permit one to query properties of the ring.
The type of a raw value is RingBase::RawValue, which helpfully
abbreviates to RawValue inside the namespace of RingBase. Wherever
possible the concrete implementations should be "exception safe", i.e. they
should offer either the strong exception guarantee or the no-throw
guarantee (according to the definitions in "Exceptional C++" by Sutter).
Bugs, Shortcomings and other ideas
Shouldn't there be hints and notes about creating your own new type of
I have chosen not to use [operator^] for computing powers because of a
significant risk of misunderstanding between programmer and compiler.
The syntax/grammar of C++ cannot be changed, and [operator^] binds less
tightly than (binary) [operator*], so any expression of the form [a*b^c]
will be parsed as [(a*b)^c]; this is almost certainly not what the
programmer intended. To avoid such problems of misunderstanding I
have preferred not to define [operator^]; it seems too dangerous.
How to swap [RefRingElem]s? Must be careful when trying to swap a
[RefRingElem] with a [RingElem] to avoid possible orphans (memory
leaks) or doubly owned values.
Note about comparison operators (<,<=,>,>=, and !=). The C++ STL
does have templates which will define all the relational operators
efficiently assuming the existence of [operator<] and [operator==].
These are defined in the namespace [std::rel_ops] in the standard
header file <utility>. I have chosen NOT to use these because they can
define only "homogeneous" comparisons; so the comparisons between
[ConstRefRingElem] and [int] or [ZZ] would still have to be written out
manually, and I prefer the symmetry of writing them all out.
See p.69ff of Josuttis for details.
Printing rings is unsatisfactory. Need a mechanism for specifying
a print name for a ring; and also a mechanism for printing out the full
definition of the ring avoiding all/some print names. For instance,
R = Q(x), S = R[a,b]; S could print as S, R[a,b] or Q(x)[a,b]. We
should allow at least the first and the last of these possibilities.
The function [myAssignZero] was NECESSARY because [myAssign(x, 0)] was
ambiguous (ambiguated by the assignment from an [mpz_t]). It is no longer
necessary, but I prefer to keep it (for the time being).
The requirement to use the type [ConstRefRingElem] for function arguments
(which should normally be [const RingElem&] is not ideal, but it seems hard
to find a better way. It is not nice to expect users to use a funny type
for their function arguments. How else could I implement access to
coefficients in a polynomial via an iterator, or access to matrix elements?
Would we want ++ and -- operators for RingElems???
I had hoped to place the auto_ptrs [myZeroPtr] and [myOnePtr] in RingBase, but
this caused trouble when rings are destroyed: the concrete ring was
destroyed before the zero and one elements were destroyed. It seems much
safer simply to duplicate the code for each ring implementation class.
Should (some of) the query functions return [bool3] values?
What about properties which are hard to determine?
There used to be an [IsFinite(R)] function in the documentation, but it
did not exist in the code. Should it exist?
How to generate random elements from a ring?
Note the slightly unusual return types for [operator+=] etc;
it seemed daft to have a reference-to-reference. In contrast, I left
the return type for [RingElem::operator=] as [RingElem&]; it felt
very odd thinking of some other type [RefRingElem] as the return type.
The dtor for [ConstRefRingElem] is deliberately not virtual; idem for
[RefRingElem]. This could potentially cause trouble if you convert a
pointer to [RingElem] into a pointer to [RefRingElem]; but if you do that,
you'd better be doubly sure about what you're doing anyway.
Some very old notes about implementing rings
This all needs to be sorted out!
Mapping elements between rings automatically
How to decide whether a value can be mapped into the current_ring?
If the rings are marked as being equivalent isomorphically then we
can just use the obvious isomorphism. A more interesting case is
when a value resides in a ring which is a natural subring of the
current_ring e.g. Z inside Q(sqrt(2))[x,y,z].
One could argue that to create Q(sqrt(2))[x,y,z] we had to follow this path
Z --> fraction field Q
Q --> polynomial ring Q[gensym], or DUP extension Q[gensym]
Q[gensym] --> quotient by gensym^2-2 to get Q(sqrt(2))
Q(sqrt(2)) --> polynomial ring (3 vars) Q(sqrt(2))[x,y,z]
From this it ought to be easy to identify natural embeddings of
Z, Q, and (possibly) Q(sqrt(2)) in Q(sqrt(2))[x,y,z]. We do not
get an embedding for Q[gensym] since we had to generate the symbol
"gensym" and no one else can create the same gensym. Because of
this it is not altogether clear that an independently created copy
of Q(sqrt(2)) can be embedded automatically, since that copy would
have a different symbol/gensym. Now if the algebraic extension were
Would we want Q[x]/(x^2-2) to be regarded as isomorphically equivalent
to Q[y]/(y^2-2)? In fact there are two possible isoms: x <---> y
and x <---> -y. I think that these should not be viewed as isom
automatically, especially as there is more than one possible choice.
In contrast, if R = Q[x]/(x^2-2), and S = Q[x]/(36-18x^2), and
T = Q[x]/(x^2-2). It is clear that Q[x] can be mapped into each
of R, S and T in a natural way. Of course, in each case x stands
for sqrt(2), and it wouldn't be too hard to spot that R and T are
"identical"; it is not quite as simple to see that R and S are
isom. Presumably with a little more effort one could create examples
where it could be jolly hard to spot that two such rings are just
the same ring. For this reason, I think no attempt should be made
to spot such "natural isoms" between "independent" rings. Had T
been created from R (e.g. by making copy via assignment) then they
would no longer be independent, and a natural isom could be deduced
automatically. Now I think about it, a facility to make a copy of
a ring WITHOUT the natural isom should be made available.
There is also a need for a way to specify that one ring embeds
naturally into another (and via which homomorphism), or indeed that
they are isomorphic. Isomorphism could be expressed by giving two
inverse homs -- the system could then check that the homs are inverse
on the generators, how it would check that the maps are homs is not
so clear (perhaps the only maps which can be created are homs).
Oooops, this would allow one to declare that Z and Q (or Z[x] and Q[x])
are isom..... need to think more about this!
A similar mechanism will be needed for modules (and vector spaces).
A module should naturally embed into a vector space over the fraction
field of the base ring....
Conceivably someone might want to change the natural embedding between
two rings. So a means of finding out what the natural embedding is
will be necessary, and also a way replacing it.
There is also a general question of retracting values into "subrings".
Suppose I have computed 2 in Q(x), can I get the integer 2 from this?
In this case I think the user must indicate explicitly that a retraction
is to occur. Obviously retraction can only be into rings "on the way"
to where the value currently resides.
Other points to note:
Q(x) = Z(x) = FrF(Z[x]) == FrF(FrF(Z)[x])
Q(alpha) = FF(Z[alpha]) though denoms in Q(alpha) can be taken in Z
Q[alpha]/I_alpha = FF(Z[alpha]/I_alpha) **BUT** the ideal on LHS
is an ideal inside Q[alpha] whereas that in RHS is in Z[alpha].
Furthermore Z[alpha]/I_alpha is "hairy" if the min poly of alpha is not monic!