2000
DOI: 10.1017/s1446788700002470
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Sums of squares of integral linear forms

Abstract: In this paper we prove that every positive definite n-ary integral quadratic form with 12 < n < 13 (respectively 14 < n < 20) that can be represented by a sum of squares of integral linear forms is represented by a sum of 2 • 3" + n + 6 (respectively 3 • 4" + n + 3) squares. We also prove that every positive definite 7-ary integral quadratic form that can be represented by a sum of squares is represented by a sum of 25 squares.

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Cited by 7 publications
(3 citation statements)
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“…Ko [18] conjectured that g Z (6) = 9, but this is disproved almost sixty years later by Kim-Oh [12] who show that g Z ( 6) is actually equal to 10. This is the last known exact value of g Z (n), although explicit upper bounds for g Z (n) for n ≤ 20 have been found; see [13] and [29].…”
Section: Introductionmentioning
confidence: 83%
“…Ko [18] conjectured that g Z (6) = 9, but this is disproved almost sixty years later by Kim-Oh [12] who show that g Z ( 6) is actually equal to 10. This is the last known exact value of g Z (n), although explicit upper bounds for g Z (n) for n ≤ 20 have been found; see [13] and [29].…”
Section: Introductionmentioning
confidence: 83%
“…Much effort went into obtaining upper and lower bounds for g Z (r). For 7 ≤ r ≤ 20, explicit bounds are known [KO2,Sa1]. The upper bounds valid for all r improved gradually: from functions growing faster than exponentially [Ic2] through an exponential [KO3] to the currently best g Z (r) = O(e (4+2 √ 2+ε) √ r ) due to Beli, Chan, Icaza and Liu [BCIL].…”
Section: Introductionmentioning
confidence: 99%
“…Much effort went into obtaining upper and lower bounds for g Z (r). For 7 ≤ r ≤ 20, explicit bounds are known [KO2,Sa1]. The upper bounds valid for all r improved gradually: from functions growing faster than exponentially [Ic2] through an exponential [KO3] to the currently best g Z (n) = O(e (4+2 and for orders of number fields [Kr, KRS, Pe, Ti], and, in an influential paper [CDLR], for affine and local algebras.…”
Section: Introductionmentioning
confidence: 99%