Solutions To Ma3417 Homework Assignment 2 Page 3
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1
2
32
2
2
2
5. Suppose that we would like to minimize (a
c)
+ (b
d)
given that ab = 4 and c
+ 4d
= 4. In more geometric
2
2
2
terms, we would like to find the minimal distance between the hyperbola xy = 4 and the ellipse x
+ 4y
= 4 in R
.
(a) Using Lagrange multipliers, write down the system of polynomial equations that one has to solve to find that
minimum.
(b) Use Gr¨ o bner bases to eliminate unknowns and rewrite your system of equations in a way suitable for solving by
extension.
Solution. (a) The Lagrange multipliers method suggests that we should look at the critical points of the function
2
2
2
2
F (a, b, c, d, α, β) = (a
+ (b
4)
+ 4d
4).
c)
d)
α(ab
β(c
We have
∂
F = 2(a
c)
αb,
a
∂
F = 2(b
d)
αa,
b
F = 2(c
2βc,
∂
a)
c
∂
F = 2(d
b)
8βd,
d
F = ab
4,
∂
α
2
2
∂
F = c
+ 4d
4.
β
This is precisely the system of equations we are aiming to solve.
(b) Computing a Gr¨ o bner basis for the LEX ordering with α > β > a > b > c > d using Singular, we get a Gr¨ o bner
basis
16
14
12
10
8
6
4
2
81d
324d
+ 3310d
5796d
+ 13653d
9004d
+ 2484d
324d
+ 16,
15
13
11
9
188352c + 4742469d
18574191d
+ 192181159d
323074301d
+
7
5
3
+ 769803276d
459193664d
+ 97662148d
7408656d,
15
13
11
53680320b
1638275463d
+ 6225285213d
65700922861d
+
9
7
5
3
+ 104081012935d
255308307124d
+ 131039721440d
24029768972d
+ 1779004976d,
15
13
11
71573760a
490312683d
+ 1953288513d
20013783641d
+
9
7
5
3
+ 34789074995d
82428604004d
+ 53519167360d
15225418492d
+ 1552217776d,
14
12
10
8
214721280β + 1638275463d
6225285213d
+ 65700922861d
104081012935d
+
6
4
2
+ 255308307124d
131039721440d
+ 24029768972d
1832685296,
14
12
10
8
107360640α + 1679218371d
6538493421d
+ 67910186537d
112888398095d
+
6
4
2
+ 270453063428d
157077454960d
+ 32607369964d
2435055472.
We note that the first equation is an equation in d only, and the further equations allow to reconstruct other
unknowns uniquely, using d.
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