$g$-invariant on unary Hermitian lattices over imaginary quadratic fields with class number $2$ or $3$

Liu, Jingbo

doi:10.48550/arxiv.2111.10825

Cited by 2 publications

(4 citation statements)

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“…While the definition of g(r) may seem more straightforward than the definition of G(r), we consider the latter to be the more useful and natural invariant. Perhaps it is not found in the literature simply because almost nothing nontrivial was known for any field of a class number other than 1 (see [Li1] for an example for Hermitian lattices). Now we can partly remedy this by deciding the G(2)-invariant of most real quadratic fields -that is, we prove Theorem 1.4.…”

Section: G(2) For Quadratic Rings Of Integersmentioning

confidence: 99%

“…However, there is a natural alternative: Let G O K (r) be obtained by replacing quadratic forms with quadratic lattices and suitably rephrasing the condition about a sum of squares of linear forms. (The precise formulation is in Definition 2.1; note that the analogous definition for Hermitian lattices is used in [BCIL,Li1,Li2].) A more encompassing result of this paper is the following:…”

Section: Introductionmentioning

confidence: 99%

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On quadratic Waring’s problem in totally real number fields

Krásenský

Yatsyna

2023

Proc. Amer. Math. Soc.

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We improve the bound of the g g -invariant of the ring of integers of a totally real number field, where the g g -invariant g ( r ) g(r) is the smallest number of squares of linear forms in r r variables that is required to represent all the quadratic forms of rank r r that are representable by the sum of squares. Specifically, we prove that the g O K ( r ) g_{\mathcal {O}_K}(r) of the ring of integers O K \mathcal {O}_K of a totally real number field K K is at most g Z ( [ K : Q ] r ) g_{\mathbb {Z}}([K:\mathbb {Q}]r) . Moreover, it can also be bounded by g O F ( [ K : F ] r + 1 ) g_{\mathcal {O}_F}([K:F]r+1) for any subfield F F of K K . This yields a subexponential upper bound for g ( r ) g(r) of each ring of integers (even if the class number is not 1 1 ). Further, we obtain a more general inequality for the lattice version G ( r ) G(r) of the invariant and apply it to determine the value of G ( 2 ) G(2) for all but one real quadratic field.

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Section: G(2) For Quadratic Rings Of Integersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On quadratic Waring’s problem in totally real number fields

Krásenský

Yatsyna

2023

Proc. Amer. Math. Soc.

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“…While the Pythagoras number is the more natural invariant than G(1), since a number is a more important object than a unary quadratic lattice, for lattices of higher rank, we consider G(r) to be the more general and natural invariant than g(r). Perhaps it is not found in the literature because almost nothing nontrivial was known for any field of a class number other than one (see [Li1] for an example for Hermitian lattices). Now we can remedy this by deciding the G(2)-invariant of most real quadratic fields -that is, we prove Theorem 1.4.…”

Section: Proof Of Theorem 41 (2)mentioning

confidence: 99%

“…For number fields with class number larger than one, the g-invariant is still well-defined, but there is a natural alternative notion: Let G O K (r) be obtained by replacing quadratic forms by quadratic lattices and suitably rephrasing the condition about a sum of squares of linear forms. (The precise formulation is in Definition 2.2; note that the analogy for Hermitian lattices already appeared in [BCIL], [Li1] and [Li2].) The main result of this paper is the following:…”

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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$g$-invariant on unary Hermitian lattices over imaginary quadratic fields with class number $2$ or $3$

Cited by 2 publications

References 13 publications

On quadratic Waring’s problem in totally real number fields

On quadratic Waring’s problem in totally real number fields

On quadratic Waring's problem in totally real number fields

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