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Polynomial parametrization of Pythagorean tuples. (English) Zbl 1237.11015

A Pythagorean \((k,l)\)-tuple over a commutative ring \(A\) is a vector \((x_1,\dots,x_{k+l}) \in A^{k+l}\) such that \(x_1^2 + \dots + x_k^2 = x_{k+1}^2 + \dots + x_{k+l}^2\). In the present paper, the authors give a polynomial parametrization of Pythagorean \((k,l)\)-tuples over the ring \(F[t]\) (where \(F\) is a field), for \(l\geq 2\). They also give the solutions for the cases \(l=1\) and \(k=2,3,4,5,9\).

MSC:

11D09 Quadratic and bilinear Diophantine equations
11E12 Quadratic forms over global rings and fields
11E25 Sums of squares and representations by other particular quadratic forms
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References:

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