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42 CHAPTER 2. POLYNOMIALS
common factors An example of this would be
x 2 − 2x + 1
x 2 − 1
versus
x − 1
x + 1 .
If we put the difference between the two over a common denominator, we
0
get
x 2 −1 = 0. The reader may complain that x2 −2x+1
x 2 −1
“is undefined when
x = 1”, whereas x−1
x+1
“has the value 0”. However, we have not defined
what we mean by such substitutions, and for the purposes of this chapter,
we are concerned with algebraic equality in the sense of proposition 11.
2.2.1 Candidness of rational functions
We have already given (Definition 5) an abstract definition of candidness, which
can also be described as “what you see is what you’ve got” mathematically.
What would this mean for rational functions (and therefore for polynomials)?
[Sto11, p.869] gives the following as a sufficient set of conditions 21 .
1. there are no compound ratios such as
x + x + 1
1 + 1 x
(2.7)
(note that this is 2x, and therefore “only” a polynomial, so violates the
general definition of candidness),
2. all ratios that occur are reduced (therefore preferring x 999 + · · · + x + 1 to
x 1000 −1
x−1
),
3. the factors and terms are ordered in an easily discerned traditional way,
such as lexically by descending degree,
4. all manifestly similar factors and terms are collected,
5. for each variable, the actual degree of every variable in a reduced ratio of
an expanded numerator and denominator would be no less than what a
user would predict assuming no cancellations. For example, assuming no
cancellations, we can predict that at most the degree of x will be 3 in the
denominator and 6 in the numerator when
x 3 + 1
x 2 − 1 + 1
x + 2
(2.8)
is reduced over a common denominator. Those are the resulting degrees,
so (2.8) is a candid representation, even though it’s probably not one of a
class of canonical representations. Conversely (2.7) violates this condition,
since we would predict it to be the ratio of a degree 2 and a degree 1
polynomial.
21 He also remarks “There can be other candid forms for rational expressions, including
[appropriately reduced, ruling out (2.7)] continued fractions. However, the complexity of
implementing a candid simplifier increases with the permissiveness of the allowed result forms.”