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16 CHAPTER 1. INTRODUCTION
[BCM94], again with a strong emphasis on genericity. Another popular system
is GAP [BL98], whose ‘kernel+library’ design was consciously [Neu95] inspired
by Maple.
The reader may well ask ‘why two different schools of thought?’ The author
has often asked himself the same question. The difference seems one of
mathematical attitude, if one can call it that. The designer of a calculus system
envisages it being used to compute an integral, factor a polynomial, multiply
two matrices, or otherwise operate on a mathematical datum. The designer of
a group theory system, while he will permit the user to multiply, say, permutations
or matrices, does not regard this as the object of the system: rather the
object is to manipulate whole groups (etc.) of permutations (or matrices, or
. . .), i.e. a mathematical structure.
1.1.3 A synthesis?
While it is too early to say that the division has been erased, it can be seen
that MAGMA, for example, while firmly rooted in the group-theory tradition,
has many more ‘calculus like’ features. Conversely, the interest in polynomial
ideals, as in Proposition 28, means that systems specialising in this direction,
such as SINGULAR [Sch03] or COCOA [GN90], use polynomial algorithms, but
the items of interest are the mathematical structures such as ideals rather than
the individual polynomials.
1.2 Expansion and Simplification
These two words, or the corresponding verbs ‘expand’ and ‘simplify’ are much
used (abused?) in computer algebra, and a preliminary explanation may be
in order. Computers of course, do not deal in mathematical objects, but,
ultimately, in certain bit patterns which are representations of mathematical
objects. Just as in ink on paper, a mathematical object may have many representations,
just as most of us would say that x + y and y + x are the same
mathematical object.
Definition 2 A correspondence f between a class O of objects and a class R of
representations is a representation of O by R if each element of O corresponds
to one or more elements of R (otherwise it is not represented) and each element
of R corresponds to one and only one element of O (otherwise we do not know
which element of O is represented). In other words “is represented by”, is the
inverse of a surjective function f or “represents” from a subset of R (the “legal
representations”) to O.
Notation 2 When discussing the difference between abstract objects and computer
representations, we will use mathematical fonts, such as x, for the abstract
objects and typewriter font, such as x, for the representations.
Hence we could represent any mathematical objects, such as polynomials, by
well-formed strings of indeterminates, numbers and the symbols +,-,*,(,).