Pythagoras numbers of orders in biquadratic fields

被引：7

作者：

Krasensky, Jakub ^{[1
]}

Raska, Martin ^{[1
]}

Sgallova, Ester ^{[1
]}

机构：

[1] Charles Univ Prague, Fac Math & Phys, Dept Algebra, Sokolovska 83, Prague 8, Czech Republic

来源：

EXPOSITIONES MATHEMATICAE | 2022年 / 40卷 / 04期

关键词：

Sum of squares; Pythagoras number; Biquadratic number field; Ring of integers; UNIVERSAL QUADRATIC-FORMS; SUMS; SQUARES;

D O I：

10.1016/j.exmath.2022.06.002

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We examine the Pythagoras number P(OK) of the ring of integers OK in a totally real biquadratic number field K. We show that the known upper bound 7 is attained in a large ./ and natural infinite family of such fields. In contrast, for almost all fields Q( 5, ./s) we prove P(OK) = 5. Further we show that 5 is a lower bound for all but seven fields K and 6 is a lower bound in an asymptotic sense.(c) 2022 Elsevier GmbH. All rights reserved.

引用

页码：1181 / 1228

页数：48

共 39 条

[1] SUMS OF SQUARES IN ALGEBRAIC FUNCTION FIELDS OVER A COMPLETE DISCRETELY VALUED FIELD [J].

Becher, Karim Johannes ;

Grimm, David ;

Van Geel, Jan .

PACIFIC JOURNAL OF MATHEMATICS, 2014, 267 (02) :257-276

[2] The length and other invariants of a real field [J].

Becher, Karim Johannes ;

Leep, David B. .

MATHEMATISCHE ZEITSCHRIFT, 2011, 269 (1-2) :235-252

[3] Sums of squares in function fields of hyperelliptic curves [J].

Becher, Karim Johannes ;

Van Geel, Jan .

MATHEMATISCHE ZEITSCHRIFT, 2009, 261 (04) :829-844

[4] Universal quadratic forms and indecomposables over biquadratic fields [J].

Cech, Martin ;

Lachman, Dominik ;

Svoboda, Josef ;

Tinkova, Magdalena ;

Zemkova, Kristysma .

MATHEMATISCHE NACHRICHTEN, 2019, 292 (03) :540-555

[5] HERMITE REDUCTION AND A WARING'S PROBLEM FOR INTEGRAL QUADRATIC FORMS OVER NUMBER FIELDS [J].

Chan, Wai Kiu ;

Icaza, Maria Ines .

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 2021, 374 (04) :2967-2985

[6] Sums of squares in rings of integers with 2 inverted [J].

Collinet, Gael .

ACTA ARITHMETICA, 2016, 173 (04) :383-390

[7]

Dzewas, 1960, MATH NACHR, V21, P233, DOI [10.1002/mana.19600210309, DOI 10.1002/MANA.19600210309]

[8]

He ZL, 2022, Arxiv, DOI arXiv:2204.10468

[9] Pythagoras numbers of fields [J].

Hoffmann, DW .

JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY, 1999, 12 (03) :839-848

[10] The Pythagoras number and the u-invariant of Laurent series fields in several variables [J].

Hu, Yong .

JOURNAL OF ALGEBRA, 2015, 426 :243-258

← 1 2 3 4 →