Composition of Sums of Squares with Integer Coefficients

Paul Yiu
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Paul Yiu: Florida Atlantic University, Department of Mathematics

A chapter in Deformations of Mathematical Structures II, 1994, pp 7-100 from Springer

Abstract: Abstract This paper is on the construction of shortest composition formulae of the form 1.1.1 $$ \left( {x_1^2 + \cdots + x_r^2} \right)\left( {y_1^2 + \cdots + y_s^2} \right) = z_1^2 + \cdots + z_n^2 $$ in which z 1 ,..., z n are polynomials in x 1 ,..., x r , y 1 ,..., y s with integer coefficients. We shall call an identity of this form an [r, s,n]ℤ formula. Composition formulae are generalizations of the classical 2–, 4–, 8–square identities which express the multiplicative property of the norms of complex numbers, quaternions and octonians respectively. The impossibility of a 16–square identity, viz. a [16,16,16] ℤ formula, was discovered in the late 1840’s. This suggested the problem of determining, for given r and s, the smallest integer n, denoted r *ℤ s, for which there exists an [r, s, n] ℤ formula. The purpose of this paper is to determine the precise values of r *ℤ s in the range 10 ≤ r, s ≤ 16.

Keywords: Left Corner; Distinct Color; Relevant Range; Composition Algebra; Composition Formula (search for similar items in EconPapers)
Date: 1994
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-011-1896-5_2

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DOI: 10.1007/978-94-011-1896-5_2

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