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82 CHAPTER 3. POLYNOMIAL EQUATIONS
Definition 47 A polynomial is said to be homogeneous if every term has the
same total degree. A set of polynomials is said to be homogeneous if each of
them separately is homogeneous. Note that we are not insisting that all terms
in the set have the same degree, merely that within each polynomial they have
the same total degree.
Definition 48 If f = ∑ ∏ n
c i j=1 xai,j j ∈ K[x 1 , . . . , x n ] is not homogeneous, and
has total degree d, we can define its homogenisation to be f 0 ∈ K[x 0 , x 1 , . . . , x n ]
as
f 0 = ∑ c i x d−∑ n
j=1 ai,j
0
n∏
j=1
x ai,j
j .
Proposition 35 If f and g are homogeneous, so is S(f, g) and h where f → g h.
Corollary 7 If the input to Algorithm 8 is a set of homogeneous polynomials,
then the entire computation is carried out with homogeneous polynomials.
The normal selection strategy is observed to work well with homogeneous polynomials,
but can sometimes be very poor on non-homogeneous polynomials.
Hence [GMN + 91] introduced the following concept.
Definition 49 The ‘sugar’ S f of a polynomial f in Algorithm 8 is defined inductively
as follows:
1. For an input f ∈ G 0 , S f is the total degree of f (even if we are working
in a lexicographic ordering)
2. If t is a term, S tf = deg(t) + S f ;
3. S f+g = max(S f , S g ).
We define the sugar of a pair of polynomials to be the sugar of their S-polynomial,
i.e. (the notation is not optimal here!) S (f,g) = S S(f,g) .
The sugar of a polynomial is then the degree it would have had we homogenised
all the polynomials before starting Algorithm 8.
Definition 50 We say that an implementation of Algorithm 8 follows a sugar
selection strategy if, at each stage, we pick a pair (i, j) such that S (gi,g j) is
minimal.
This does not completely specify what to do, and it is usual to break ties with the
normal selection strategy (Definition 46), and “sugar then normal” is generally
just referred to as “sugar”.