Sums of squares of integral linear forms

Icaza, María Inés

doi:10.4064/aa-74-3-231-240

Cited by 16 publications

(15 citation statements)

References 14 publications

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“…^ o c gz(n) < { and g z (7) < 25. [3-4" + /i + 3 f o r l 4 < w < 2 0 , These bounds are better than those of Icaza [2] and Oh [10]. We use the terminology and notation of O'Meara [12].…”

Section: Introductionmentioning

confidence: 62%

“…In this paper we give an upper bound for gz(n) for 12 < n < 20 and for n = 7: These bounds are better than those of Icaza [2] and Oh [10]. We use the terminology and notation of O'Meara [12].…”

Section: Introductionmentioning

confidence: 99%

“…Kim and Oh [3,4] proved that every positive definite 6-ary integral quadratic form that can be represented by a sum of squares is represented by a sum of 10 squares, and showed that g z (6) = 10. [2] Sums of squares 299…”

Section: Introductionmentioning

confidence: 99%

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Sums of squares of integral linear forms

Sasaki¹

2000

J Aust Math Soc A

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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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“…^ o c gz(n) < { and g z (7) < 25. [3-4" + /i + 3 f o r l 4 < w < 2 0 , These bounds are better than those of Icaza [2] and Oh [10]. We use the terminology and notation of O'Meara [12].…”

Section: Introductionmentioning

confidence: 62%

Section: Introductionmentioning

confidence: 99%

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Sums of squares of integral linear forms

Sasaki¹

2000

J Aust Math Soc A

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“…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%

On quadratic Waring's problem in totally real number fields

Krásenský¹,

Yatsyna²

2021

Preprint

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We improve the bound of the g-invariant of the ring of integers of a totally real number field, where the g-invariant g(r) is the smallest number of squares of linear forms in r variables that is required to represent all the quadratic forms of rank r that are representable by the sum of squares. Specifically, we prove that the gO K (r) of the ring of integers OK of a totally real number field K is at most gO F ([K : F ]r) for any subfield F of K. This yields a sub-exponential upper bound for g(r) of each ring of integers (even if the class number is not 1). Further, we obtain a more general inequality for the lattice version G(r) of the invariant and apply it to determine the value of G( 2) for all but two real quadratic fields. √ 2+ε) √n ) due to Beli, Chan, Icaza and Liu [BCIL]. The g-invariant can be generalised to g R ( • ) of an arbitrary ring R in an obvious way, simply replacing forms over Z by forms over R. In particular, the value P(R) = g R (1) is the Pythagoras number of R, much examined both for arbitrary fields (see, for example, [Le])

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“…The definition of the g-invariants of Z is a special case of the g-invariants of a commutative ring A first appeared in the third author's paper [9], although the special case where A is a field is studied earlier in [1]. Using a deep result in representations of positive definite integral quadratic forms [7], the third author [9] shows that when O is the ring of integers of a totally real number field, g O (n) is bounded above by a function of n which is at least an exponential function of the class number of I n+3 as a quadratic form over O. The latter is expected to be growing extremely fast as n increases.…”

Section: Introductionmentioning

confidence: 99%

On a Waring’s problem for integral quadratic and Hermitian forms

Beli

Chan

Icaza

et al. 2018

Trans. Amer. Math. Soc.

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For each positive integer n, let g Z (n) be the smallest integer such that if an integral quadratic form in n variables can be written as a sum of squares of integral linear forms, then it can be written as a sum of g Z (n) squares of integral linear forms. We show that as n goes to infinity, the growth of g Z (n) is at most an exponential of √ n. Our result improves the best known upper bound on g Z (n) which is in the order of an exponential of n. We also define an analogous number g * O (n) for writing hermitian forms over the ring of integers O of an imaginary quadratic field as sums of norms of integral linear forms, and when the class number of the imaginary quadratic field is 1, we show that the growth of g * O (n) is at most an exponential of √ n. We also improve results of Conway-Sloane [2] and Kim-Oh [14] on s-integral lattices.

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Sums of squares of integral linear forms

Cited by 16 publications

References 14 publications

Sums of squares of integral linear forms

Sums of squares of integral linear forms

On quadratic Waring's problem in totally real number fields

On a Waring’s problem for integral quadratic and Hermitian forms

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