A cubic ring of integers with the smallest Pythagoras number

Krásenský, Jakub

doi:10.1007/s00013-021-01662-5

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2022

2023

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Cited by 4 publications

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“…In this case, obtaining an upper bound for P(R) is generally much more difficult than for P(K). The interested readers are referred to [Pfi95, Chapter 7], [KRS21], [Tin21], [Krá22] and the references therein for more information on this topic.…”

Section: Introductionmentioning

confidence: 99%

Pythagoras number of quartic orders containing $\sqrt{2}$

He¹,

Hu²

2022

Preprint

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Section: Introductionmentioning

confidence: 99%

Pythagoras number of quartic orders containing $\sqrt{2}$

He¹,

Hu²

2022

Preprint

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Pythagoras numbers of orders in biquadratic fields

Krásenský

Raška

Ester

2022

Expositiones Mathematicae

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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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A cubic ring of integers with the smallest Pythagoras number

Cited by 4 publications

References 18 publications

Pythagoras number of quartic orders containing $\sqrt{2}$

Pythagoras number of quartic orders containing $\sqrt{2}$

Pythagoras numbers of orders in biquadratic fields

On quadratic Waring’s problem in totally real number fields

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