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+6
s
1
x
30
lb
+2
s
0
x
22
b
2
+2
s
1
x
21
b
2
+2
s
2
x
20
b
2
+O(3)
,
x
b
(
l,b
)=
x
01
+
x
11
l
+2
x
02
b
+
x
21
l
2
+2
x
12
lb
+3
x
03
b
2
+
(15.48)
+
x
31
l
3
+2
x
22
l
2
b
+3
x
13
lb
2
+4
x
04
b
3
+O(4)
,
x
bb
(
l,b
)=2
x
02
+2
x
12
l
+6
x
03
b
+
(15.49)
+2
x
22
l
2
+6
x
13
lb
+12
x
04
b
2
+O(3)
,
r
b
x
b
(
l,b
)=(
r
1
+2
r
2
b
+3
r
3
b
2
+ O(3))
x
b
=
r
1
x
01
+
r
1
x
11
l
+2
r
1
x
02
b
+2
r
2
x
01
b
+
r
1
x
21
l
2
+2
r
1
x
12
lb
+
(15.50)
+2
r
2
x
11
lb
+3
r
1
x
03
b
2
+4
r
2
x
02
b
2
+3
r
3
x
01
b
2
+O(3)
,
rx
bb
(
l,b
)=(
r
0
+
r
1
b
+
r
2
b
2
+ O(3))
x
bb
=
2
r
0
x
02
+2
r
0
x
12
l
+6
r
0
x
03
b
+2
r
1
x
02
b
+2
r
0
x
22
l
2
+6
r
0
x
13
lb
+
(15.51)
2
r
1
x
12
lb
+12
r
0
x
04
b
2
+6
r
1
x
03
b
2
+2
r
2
x
02
b
2
+O(3)
.
While (
15.47
), (
15.50
), and (
15.51
) represent the polynomial solution of (
15.42
), namely for
x
(
l,b
),
a corresponding solution for (
15.43
) could be found as soon as we replace
x
and
y
, namely for the
polynomial solution
y
(
l,b
). Let us write down the
n
1 constraints for
n
+1 polynomials given by
the zero identity of the sum of the three terms of (
15.47
)(
sx
ll
,firstterm),(
15.50
)(
r
b
x
b
, second
term), and (
15.51
)(
rx
bb
, third term).
−
Corollary 15.1 (Laplace-Beltrami equations solved in the function space of bivariate polynomi-
als).
If a polynomial (
15.29
)-(
15.32
)ofdegree
n
fulfills the Laplace-Beltrami equations (
15.42
)
and (
15.43
), then there are
n
−
1 coecient constraints, namely
n
=2:
2
s
0
x
20
+2
r
0
x
02
+
r
1
x
01
=0
,
(15.52)
2
s
0
y
20
+2
r
0
y
02
+
r
1
y
01
= 0;
(15.53)
n
=3:
6
s
0
x
30
+2
r
0
x
12
+
r
1
x
11
=0
,
(15.54)
6
s
0
y
30
+2
r
0
y
12
+
r
1
y
11
=0
,
(15.55)
s
0
x
21
+
s
1
x
20
+3
r
0
x
03
+2
r
1
x
02
+
r
2
x
01
=0
,
(15.56)
s
0
y
21
+
s
1
y
20
+3
r
0
y
03
+2
r
1
y
02
+
r
2
y
01
= 0;
(15.57)
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