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2.1. WHAT ARE POLYNOMIALS? 39
p:=x+1;
q:=p;
r:=p*q;
Figure 2.2: Code fragment A — a graph
p:=x+1;
q:=x+1;
r:=p*q;
Figure 2.3: Code fragment B — a tree
We said earlier that the only known algorithms for computing greatest
common divisors were recursive. This is essentialy true, and means that
the computation of greatest common disivors of straight-line programs is
not a straight-forward process [Kal88].
Additive Complexity This [Ris85, Ris88] is similar to a straight-line program,
except that we only count the number of (binary) addition/subtraction
nodes, i.e. multiplication and exponentiation are ‘free’. Hence the
degree is unbounded in terms of the additive complexity, but for a given
expression (tree/DAG) can be bounded by a tree-walk. A univariate polynomial
of additive complexity a has at most C a2 real roots for some absolute
constant C: conjecturally this can be reduced to 3 a . These bounds
trivially translate to the straight-line program and expression tree cases.
“Additive complexity” is more of a measure of the ‘difficulty’ of a polynomial
than an actual representation. Of the others, the first was used in Macsyma for
its “general expression”, and the second is used in Maple 17 . In fact, Macsyma
would 18 allow general DAGs, but would not force them. Consider the two code
fragments in figures 2.2 and 2.3. In the case of figure 2.2, both systems would
produce the structure in figure 2.4. For figure 2.3, Macsyma would produce the
structure 19 in figure 2.5. Maple would still produce the structure of figure 2.4,
since the hashing mechanism would recognise that the two x + 1 were identical.
2.1.6 The Newton Representation
For simplicity, in this subsection we will only consider the case of charateristic
0: finite characteristic has some serious technical difficulties, and we refer the
17 Until such time as operations such as expand are used!
18 Not explicitly, but rather as a side-effect of the fact that Macsyma is implemented in Lisp,
which cares little for the difference. The basic function EQUAL does not distinguish between
acyclic and cyclic structures
19 The x is shown as shared since the Lisp implementation will store symbols unqiuely.