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3.3. NONLINEAR MULTIVARIATE EQUATIONS: DISTRIBUTED 89
H := ∅; i := j := 0
Enumerate the monomials irreducible under H in increasing order for > ′′
#When H is a Gr¨bner base, this is finite by proposition 32
#This enumeration needs to be done lazily
for each such m
Let m ∗ → G v
if v = ∑ j
k=1 c kv k
then h i+1 := m − ∑ j
k=1 c km k
H := H ∪ {h i+1 }; i := i + 1
else j := j + 1; m j := m; v j := v
return H
#It is not totally trivial that H is a Gröbner base, but it is [FGLM93].
Since this algorithm is basically doing linear algebra in the space spanned by the
irreducible monomials under G, whose dimension D is the number of solutions
(proposition 32), it is not surprising that the running time seems to be O(D 3 ),
whose worst case is O(d 3n ).
Open Problem 3 (Complexity of the FGLM Algorithm) The complexity
of the FGLM algorithm (Algorithm 11) is O(D 3 ) where D is the number
of solutions. Can faster matrix algorithms such as Strassen–Winograd [Str69,
Win71] speed this up?
An an example of the FGLM algorithm, we take the system Aux from their
paper 12 , with three polynomials
abc + a 2 bc + ab 2 c + abc 2 + ab + ac + bc
a 2 bc + a 2 b 2 c + b 2 c 2 a + abc + a + c + bc
a 2 b 2 c + a 2 b 2 c 2 + ab 2 c + ac + 1 + c + abc
The total degree Gröbner basis has fifteen polynomials, whose leading monomials
are
c 4 , bc 3 , ac 3 , b 2 c 2 , abc 2 , a 2 c 2 , b 3 c, ab 2 c, a 2 bc, a 3 c, b 4 , ab 3 , a 2 b 2 , a 3 b, a 4 .
This defines a zero-dimensional ideal (c 4 , b 4 and a 4 occur in this list), and we
can see that the irreducible monomials are
1, c, c 2 , c 3 , b, bc, bc 2 , b 2 , b 2 c, b 3 , a, ac, ac 2 , ab, abc, ab 2 , a 2 , a 2 c, a 2 b, a 3 :
twenty in number (as opposed to the 64 we would have if the basis only had
the polynomials a 4 + · · · , b 4 + · · · , c 4 + · · ·). If we wanted a purely lexicographic
base to which to apply Gianni-Kalkbrener, we would enumerate the monomials
in lexicographic order as
12 The system is obtained as they describe, except that the substitutions are x 5 = 1/c,
x 7 = 1/a.