Page 55 - Geometric Modeling and Algebraic Geometry

52 S. Breske et al.
Proof. By the preceding lemma, the number of critical points with non-zero critical
value equals the number of bounded cells of the real simple line arrangement. The
upper bound (3.6) for the maximum number M b (d) of black cells of a real simple

d
line arrangement now gives the result, because the line arrangement has exactly
2
critical points with critical value 0.
Chmutov showed a much more general result ([4], see [6] for the case of non-
degenerate critical points): For a plane curve of degree d the maximum number
d . In [6], he conjectured
of critical points on two levels does not exceed ≈ 7 2
8
d to be the actual maximum which is attained by the complex line arrange-
≈ 5 2
6
ments F A 2 (x, y) he used for his construction (and also by the real line arrangement
d
F A 2 (x, y)). Thus, our theorem 4 is the verification of Chmutov’s conjecture in the
,d
particular case of real simple line arrangements. As Chmutov remarked in [5], such
an upper bound immediately implies an upper bound on the maximum number of
nodes on a surface in separated variables:
Corollary 5. A surface of the form p(x, y)+ q(z)=0 cannot have more than
d · d + d · d =
3
≈ 1 2 1 1 2 1 5 d nodes if p(x, y) is a real simple line arrangement.
2 2 3 2 12
This number is attained by the surfaces Chm A 2 (x, y, z) defined in theorem 2.
,d
3.4 Concluding Remarks

Comparing our bound from corollary 5 to the upper bound ≈ 5 d on the zero th
3
12
Betti number (see e.g., [2] or [7]) one is tempted to ask if it is possible to deform
our singular surfaces to get examples with many real connected components. But
our surfaces Chm A 2 (x, y, z) only contain A singularities which locally look like
−
,d 1
a cone (x + y − z =0). When removing the singularities from the zero-set of the
2
2
2
surface every connected component contains at least three of the singularities. Thus,
th
the zero Betti number of a small deformation of our surfaces are not larger than
d resulting from Bihan’s construction
≈ 5 d which is far below the number ≈ 13 3
3
3·12 36
[2].
Conversely, we may ask if it is always possible to move the lines of a simple real
line arrangement in such a way that all critical points which have a critical value of
the same sign can be chosen to have the same critical value. If this were true then it
would be possible to improve our lower bound for the maximum number µ (d) of
real nodes on a real surface of degree d slightly because it is known that the upper
bounds for the maximum number M b (d) of black cells are in fact exact for infinitely
many d. E.g., in the already cited article [11], Harborth gave an explicit arrangement
of 13 straight lines which has 1 ·13 + ·13 − 13 = 47 bounded black regions.
1
2
3 3
When regarding this arrangement as a polynomial of degree d =13 it has exactly
one critical point with a negative critical value within each of the black regions. If all
these negative critical values can be chosen to be the same then such a polynomial

will lead to a surface with 13 · 13−1 +47· 13−1 = 750 > 732 nodes. Similarly,
2 2 2
such a surface of degree 9 would have 228 > 216 nodes. In the case of degree 7 the