ISSN:
1436-5081
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract x 1 2 +...+x n 2 —ax 1...x n =b. First we describe a combinatorial presentation of a group of automorphisms of this equation, ifn=3, then we getPGL (2, ℤ) as such a group of automorphisms of this equation. This gives analytical applications becausePGL (2, ℤ) acts discontinously on the set {(x 1,x 2,x 3)∣0〈x 1,x 2,x 3 andx 1 2 +x 2 2 +x 3 2 −x 1 x 2 x 3=b≤0}⊂ℝ3. Further we ask for fundamental solutions of this equation. Finally, letx 1,x 2,x 3∈ℕ withx 1 x 2 2 +x 3 2 −−x 1 x 2 x 3=≤0 Then there areA, B∈SL(2, ℝ) with trA=x 1, trB=x 2 and trA B=x 3, and the group (A, B) is a discrete free group of rank two. In analysis we are interested in the question whether there are evenA, B∈SL(2, ℚ) with trA=x 1, trB=x 2 and trA B=x 3. We give necessary and sufficient conditions for that and remark that this question is connected with the ternary quadratic formk1p 2+k2q 2−r 2,k 1=x 1 2 ,k 2=16(x 2 2 +x 1 2 +x 3 2 −x 1 x 2 x 3−4), which has some invariant properties.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01534865
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