journaltitleJournal of the American Mathematical Society
shorttitleKähler-Einstein metrics on Fano manifolds. I
titleKähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities

annotationComment: 21 pages. Corrected minor typos, included additional citations, clarified the definition of the coupled Futaki invariant
groupsSymplectic connection
journaltitleJ. Symplect. Geom.
notearXiv:1901.10454 [math] version: 2 type: article
titleOn coupled constant scalar curvature Kähler metrics

Universität Leipzig

annotationThe results of this thesis are contained in the the publications "Normal Form of Equivariant Maps in Infinite Dimensions", "Singular Symplectic Reduction In Infinite Dimension" (in preparation) and "Singular symplectic cotangent bundle reduction of gauge field theory".
groupsMy publications
titleNormal Form of Equivariant Maps and Singular Symplectic Reduction in Infinite Dimensions with Applications to Gauge Field Theory

groupsMy publications
pubstateIn preparation
titleSingular symplectic reduction in infinite dimensions

annotationThis is the first paper in the series, where we construct Lie group extensions of diffeomorphism groups using the Kostant-Souriau prequantum construction. This time we use the space of embeddings $Emb(S, M)$. Not directly used in the Research Project.
groupsMy publications
journaltitleInt. Math. Res. Not.
titleCentral extensions of Lie groups preserving a differential form

annotationThis is perhaps my best paper. We've submitted it to JAMS, where it was recently rejected. It is not directly used in the Research Project, but it shows that there are phenomena occurring in the infinite-dimensional setting: In finite dimensions, almost every action occurring in "nature" has a momentum map. In contrast, this work shows that classical momentum maps for diffeomorphism groups almost never exist.
groupsMy publications
titleGroup-valued momentum maps for actions of automorphism groups

groupsSymplectic connection
journaltitleMath. Ann.
titleThe Gromov norm of the Kähler class of symmetric domains
Symmetric Domain,Gromov Norm,paper

journaltitleProc. London Math. Soc. (3)
titleAnti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles

World Sci. Publ., River Edge, NJ

booktitleFields Medallists' Lectures
seriesWorld Sci. Ser. 20th Century Math.
titleRemarks on gauge theory, complex geometry and 4-manifold topology
57R57 (57N13 58D27 58G05)

booktitleSurv. Differ. Geom.
titleMoment maps in differential geometry

Springer International Publishing

booktitleGeometry, Analysis and Probability
groupsGIT, Symplectic connection
titleThe Ding functional, Berndtsson convexity and moment maps
Mathematics - Differential Geometry,53C55,paper

Research Institute for Mathematical Sciences, Kyoto University

groupsSymplectic connection
journaltitleCommentarii Mathematici Helvetici
notearXiv:math/0112012 version: 6 type: article
titleCommutator length of symplectomorphisms
atics - Symplectic Geometry,Mathematics - Differential Geometry,Mathematics - Group Theory,Mathematics - Geometric Topology,53D22,53D05,53D40,53D45,paper

annotationComment: 15 pages; comments are welcome!
groupsSymplectic connection
notearXiv:1605.08019 [math] type: article
titleBoltzmann's entropy and Kähler-Ricci solitons
Mathematics - Differential Geometry,paper

groupsSymplectic connection
journaltitleCommunications in Mathematical Physics
titleThe manifold of compatible almost complex structures and geometric quantization
Line Bundle,Toeplitz Operator,Symplectic Form,Parallel Transport,Geometric Quantization,paper

groupsSymplectic connection
journaltitleJ. Symplectic Geom.
titleCritical symplectic connections on surfaces

journaltitleSugaku Expositions
noteTranslation of Sûgaku 42 (1990), no. 3, 231–243
titleModuli space of polarized algebraic manifolds and Kähler metrics

annotationComment: 20 pages, to appear in Kodai Mathematical Journal
groupsSymplectic connection
journaltitleKodai Math. J.
notearXiv: math/0603706 version: 2
titleHarmonic total Chern forms and stability
atics - Differential Geometry,Mathematics - Symplectic Geometry,53C55,53C21,paper

Int. Press, Boston, MA

booktitleProceedings of the International Consortium of Chinese Mathematicians 2018
groupsSymplectic connection
titleDeformation quantization and Kähler geometry with moment map

annotationComment: The curvature condition is added in the main result, and other modifications are made
groupsSymplectic connection
journaltitleJ. Symplectic Geom.
titleCahen-Gutt moment map, closed Fedosov star product and structure of the automorphism group
Mathematics - Differential Geometry,32Q15,46L65,53C55,paper

groupsSymplectic connection
journaltitleMath. Z.
titleConformally Einstein-Maxwell Kähler metrics and structure of the automorphism group

Providence, RI: American Mathematical Society (AMS)

booktitleIntegrability, Quantization, and Geometry Ii. Quantum Theories and Algebraic Geometry. Dedicated to the Memory of Boris Dubrovin 1950–2019
isbn978-1-4704-5592-7; 978-1-4704-6435-6
titleA moment map interpretation of the Ricci form, Kähler–Einstein structures, and Teichmüller spaces

journaltitleAsian J. Math.
titleComplex structures, moment maps, and the Ricci form

groupsSymplectic connection
titleCalabi’s Extremal Kähler Metrics: An Elementary Introduction
ive.org/web/20210108083652/https://cims.nyu.edu/ rodion/lib/P.

Cham: Springer

groupsGIT, Symplectic connection
isbn978-3-030-89299-9; 978-3-030-89300-2
seriesLect. Notes Math.
titleThe Moment-Weight Inequality and the Hilbert-Mumford Criterion. GIT from the Differential Geometric Viewpoint
.ethz.ch/ salamon/PREPRINTS/momentweight.pdf
Hessian,momentum map squared,paper

pubstateTo appear
titleInfinite-dimensional Lie Groups. General Theory and Main Examples

Academic Press

subtitleDe Rham Cohomology of Manifolds and Vector Bundles
titleConnections, Curvature, and Cohomology
58-02 (55FXX 57FXX 57D20),paper

Publications CRM

subtitleAspects géométriques et algébriques, généralisations
titleL'algèbre et le groupe de Virasoro

groupsSymplectic connection
journaltitleLett. Math. Phys.
titleRemarks on symplectic connections
53D35,53D20,58J60,53C35,symplectic connection,Ricci-flat,symmetric symplectic spaces,Ricci-type connections,symplectic induction,paper

* Entry 'Gutt2006' (d:/Research/A_Central/Latex/bibtex/bib/Literature.bib): Invalid format '2019-02-21TZ' of date field 'urldate' - ignoring

groupsNash Moser
journaltitleBull. Amer. Math. Soc
titleThe inverse function theorem of Nash and Moser

journaltitleCommunications in Analysis and Geometry
titleLocalization for equivariant cohomology with varying polarization

World Scientific Publishing Co.

groupsSymplectic connection
journaltitleInt. J. Math.
titleHermitian Calabi functional in complexified orbits
Hermitian Calabi functional,extremal almost Kähler metrics,complexified orbit,Hermitian Calabi flow,Hermitian Calabi functional,extremal almost Kähler metrics,complexified orbit,Hermitian Calabi flow,paper

De Gruyter Open Access

groupsSymplectic connection
journaltitleComplex Manifolds
titleOn the transverse Scalar Curvature of a Compact Sasaki Manifold
Transverse Scalar curvature,Symmetric space,Moment map,paper

International Press of Boston

groupsSymplectic connection
journaltitleAsian J. Math.
titleKähler–Ricci soliton and H-functional
Applied Mathematics,General Mathematics,paper

Berlin, Heidelberg

booktitleDifferential Geometrical Methods in Mathematical Physics
groupsSymplectic connection
seriesLecture Notes in Math.
titleConnections on symplectic manifolds and geometric quantization
Vector Bundle,Symplectic Manifold,Geometric Quantization,Hermitian Manifold,Order Differential Operator

booktitleConference Held at Katata, September 3–8, 1979. Proceedings of the 6th International Taniguchi Symposium
howpublishedSendai/Japan: Tohoku University, Department of Mathematics. v, 23 p. (1979).
titleNon-linear problems in geometry

groupsSymplectic connection
journaltitleInt. Math. Res. Notices
titleCoupled Kähler–Einstein metrics

Berlin, Heidelberg

booktitleLecture Notes in Mathematics
groupsSymplectic connection
seriesLecture Notes in Mathematics
titleThe fundamental groups of algebraic varieties
Fundamental Group,Heisenberg Group,Massey Product,Hard Lefschetz Theorem,Compact Closed Manifold

Springer, Berlin

booktitleAlgebraic Geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978)
seriesLecture Notes in Math.
titleThe length of vectors in representation spaces
14L30 (20G05)

Princeton University Press

seriesMathematical Notes
titleCohomology of Quotients in Symplectic and Algebraic Geometry

Springer Berlin Heidelberg

titleTransformation Groups in Differential Geometry

journaltitleProc. Japan Acad. Ser. A Math. Sci.
titleCurvature and stability of vector bundles
53C55 (32L99),paper

American Mathematical Society

seriesMathematical Surveys and Monographs
titleThe Convenient Setting of Global Analysis
Infinite dimensional manifolds,paper

groupsSymplectic connection
journaltitleAnn. Global Anal. Geom.
titleInfinite dimensional moment map geometry and closed Fedosov’s star products

groupsSymplectic connection
journaltitleJ. Symplectic Geom.
titleFutaki invariant for Fedosov's star products

groupsSymplectic connection
journaltitleJ. Geom. Anal.
titleAutomorphisms and deformations of conformally Kähler, Einstein-Maxwell metrics
Conformally Kähler,Einstein–Maxwell metric,Group of holomorphic automorphisms,Licherowicz operator,53C56,paper

annotationComment: 19 pages
groupsSymplectic connection
titleMoment map, convex function and extremal point
Mathematics - Differential Geometry,Mathematics - Algebraic Geometry,Mathematics - Symplectic Geometry,paper

groupsSymplectic connection
journaltitleInt. J. Math.
titleExtremal almost-Kähler metrics
Mathematics - Differential Geometry,53B35,paper

groupsSymplectic connection
journaltitleMathematische Zeitschrift
titleExtremal K-contact metrics
Moment map,K-Contact structure,Extremal metrics,Futaki-invariant,53D10,53B35,53D20,paper

Tohoku University, Mathematical Institute

groupsSymplectic connection
issn0040-8735, 2186-585X
journaltitleTohoku Mathematical Journal
titleKähler-Einstein metrics for manifolds with nonvanishing Futaki character

groupsSymplectic connection
journaltitleProc. Roy. Soc. Edinburgh Sect. A
titleSymplectic connections and the linearisation of Hamiltonian systems
58F05 (35F20 53C05),paper

Nagoya Mathematical Journal

journaltitleNagoya Mathematical Journal
titleSur la structure du groupe d'homéomorphismes analytiques d'une certaine variété káhlérienne

Univ. Paris 7

groupsSymplectic connection
titleSymplectomorphismes hamiltoniens et métriques kählériennes

groupsSymplectic connection
titleH-functional and Matsushima type decomposition theorem
Hessian,momentum map squared,paper

American Mathematical Society (AMS)

groupsSymplectic connection
journaltitleProc. Amer. Math. Soc.
titleHessian of the Ricci-Calabi functional
53C55 (53C25 58E11),paper

groupsSymplectic connection
journaltitleAnnals of Global Analysis and Geometry
titleCalabi type functionals for coupled Kähler–Einstein metrics
Coupled Kähler–Einstein metric,Coupled Ding functional,Matsushima type decomposition theorem,Primary 53C25,Secondary 53C55,58E11,paper

journaltitleMonatsh. Math.
titleFlux homomorphisms and principal bundles over infinite dimensional manifolds
2000 Mathematics Subject Classification: 22E65; 58B20,53D50; Key words: Flux homomorphism,symplectic manifold,holonomy group,symplectomorphism,central extension,paper

Springer Japan

journaltitleJapan. J. Math.
titleTowards a Lie theory of locally convex groups

Birkhäuser Boston

titleMomentum Maps and Hamiltonian Reduction

journaltitleCompositio Mathematica
titleFormules de localisation en cohomologie equivariante
53D20 (55N91 57S15),paper

Elsevier BV

titleThe moment map and equivariant cohomology with generalized coefficients
53D20 (57S15 58A12),paper

journaltitleJournal of the London Mathematical Society
titleThe energy flow on the loop space of a compact lie group

groupsSymplectic connection
journaltitleSelecta Mathematica
titleContinuous cohomology of the group of volume-preserving and symplectic diffeomorphisms, measurable transfer and higher asymptotic cycles
Continuous cohomology,symplectomorphism groups.,paper

groupsSymplectic connection
journaltitleActa Mathematica Hungarica
titleOn products of unbounded operators

European Mathematical Society - EMS - Publishing House GmbH

annotationComment: 40 pages; minor changes; submitted
groupsSymplectic connection
journaltitleCommentarii Mathematici Helvetici
notearXiv: 1105.5814
titleThe action homomorphism, quasimorphisms and moment maps on the space of compatible almost complex structures
Mathematics - Symplectic Geometry,Mathematics - Differential Geometry,Mathematics - Group Theory,paper

* Entry 'Shelukhin2014' (d:/Research/A_Central/Latex/bibtex/bib/Literature.bib): Invalid format '2020-03-12TZ' of date field 'urldate' - ignoring

annotationComment: 47 pages
journaltitleCommun. Pure Appl. Math.
notearXiv:1211.4669 [math] type: article
titleK-Stability and Kähler-Einstein metrics
Mathematics - Differential Geometry,Mathematics - Algebraic Geometry,Mathematics - Analysis of PDEs,53 Differential geometry,14 Algebraic geometry,paper

Institut Mittag-Leffler

groupsSymplectic connection
issn0001-5962, 1871-2509
journaltitleActa Mathematica
titleUniqueness of Kähler-Ricci solitons

groupsSymplectic connection
journaltitleComment. Math. Helv.
titleAffine Zusammenhänge auf Mannigfaltigkeiten mit fast-symplektischer Struktur

Wiley Subscription Services, Inc., A Wiley Company

groupsAnalytic questions
journaltitleCommun. Pure Appl. Math.
titleOn the existence of Hermitian-Yang-Mills connections in stable vector bundles

International Press of Boston

groupsSymplectic connection
journaltitleJournal of Symplectic Geometry
titleAlmost-Kähler smoothings of compact complex surfaces with $A_1$ singularities
53C55 (57R18),paper

journaltitleJ. Lie Theory
titleCentral extensions of the Lie algebra of symplectic vector fields

groupsSymplectic connection
journaltitleAnn. Global Anal. Geom.
titleHessians of the Calabi functional and the norm function
Hessian,momentum map squared,paper

* Entry 'Wang2006' (d:/Research/A_Central/Latex/bibtex/bib/Literature.bib): Invalid format '2020-02-20TZ' of date field 'urldate' - ignoring

International Press of Boston

groupsSymplectic connection
journaltitleComm. Anal. Geom.
titleMoment map, Futaki invariant and stability of projective manifolds
Hessian,momentum map squared,paper

* Entry 'Wang2004' (d:/Research/A_Central/Latex/bibtex/bib/Literature.bib): Invalid format 'CURRENT_TIMESTAMP' of date field 'urldate' - ignoring

journaltitleJournal of Geometry and Physics
titleTwo-dimensional gauge theories revisited
58D27 (14D20 58D29 81T13 81T40),paper

annotationComment: 33 pages. Corrections and a few paragraphs re-written. To appear in Jour. Sympl. Geom
journaltitleJournal of Symplectic Geometry
titleLocalization for the norm-square of the moment map and the two-dimensional Yang-Mills integral
omentum map squared,paper

journaltitleThe Annals of Mathematics
titleOn unitary ray representations of continuous groups

Nagoya Mathematical Journal

journaltitleNagoya Mathematical Journal
titleSur la structure du groupe d'homéomorphismes analytiques d'une certaine variété káhlérienne

groupsSymplectic connection
journaltitleComment. Math. Helv.
titleAffine Zusammenhänge auf Mannigfaltigkeiten mit fast-symplektischer Struktur

Academic Press

subtitleDe Rham Cohomology of Manifolds and Vector Bundles
titleConnections, Curvature, and Cohomology
58-02 (55FXX 57FXX 57D20),paper

Springer Berlin Heidelberg

titleTransformation Groups in Differential Geometry

Fondation L'Enseignement Mathématique

journaltitleEnseign. Math. (2)
titleOn the characteristic classes of groups of diffeomorphisms

booktitleConference Held at Katata, September 3–8, 1979. Proceedings of the 6th International Taniguchi Symposium
howpublishedSendai/Japan: Tohoku University, Department of Mathematics. v, 23 p. (1979).
titleNon-linear problems in geometry

Springer, Berlin

booktitleAlgebraic Geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978)
seriesLecture Notes in Math.
titleThe length of vectors in representation spaces
14L30 (20G05)

Berlin, Heidelberg

booktitleDifferential Geometrical Methods in Mathematical Physics
groupsSymplectic connection
seriesLecture Notes in Math.
titleConnections on symplectic manifolds and geometric quantization
Vector Bundle,Symplectic Manifold,Geometric Quantization,Hermitian Manifold,Order Differential Operator

Princeton Univ. Press, Princeton, N.J.

booktitleSeminar on Differential Geometry
seriesAnn. of Math. Stud.
titleExtremal Kähler metrics
53C55 (58E30)

groupsNash Moser
journaltitleBull. Amer. Math. Soc
titleThe inverse function theorem of Nash and Moser

journaltitleProc. Japan Acad. Ser. A Math. Sci.
titleCurvature and stability of vector bundles
53C55 (32L99),paper

journaltitleJournal of the London Mathematical Society
titleThe energy flow on the loop space of a compact lie group

journaltitlePhilos. Trans. Roy. Soc. London Ser. A
titleThe Yang-Mills equations over Riemann surfaces

Birkhäuser Boston

booktitleArithmetic and Geometry
titleConvexity and Loop Groups

Princeton University Press

seriesMathematical Notes
titleCohomology of Quotients in Symplectic and Algebraic Geometry

Springer, Berlin

booktitleDifferential Geometry and Complex Analysis
groupsSymplectic connection
titleExtremal Kähler metrics II
53C55 (58E30)

journaltitleProc. London Math. Soc. (3)
titleAnti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles

Wiley Subscription Services, Inc., A Wiley Company

groupsAnalytic questions
journaltitleCommun. Pure Appl. Math.
titleOn the existence of Hermitian-Yang-Mills connections in stable vector bundles

groupsSymplectic connection
journaltitleMath. Ann.
titleThe Gromov norm of the Kähler class of symmetric domains
Symmetric Domain,Gromov Norm,paper

Berlin, Heidelberg

booktitleLecture Notes in Mathematics
groupsSymplectic connection
seriesLecture Notes in Mathematics
titleThe fundamental groups of algebraic varieties
Fundamental Group,Heisenberg Group,Massey Product,Hard Lefschetz Theorem,Compact Closed Manifold

groupsSymplectic connection
journaltitleProc. Roy. Soc. Edinburgh Sect. A
titleSymplectic connections and the linearisation of Hamiltonian systems
58F05 (35F20 53C05),paper

journaltitleSugaku Expositions
noteTranslation of Sûgaku 42 (1990), no. 3, 231–243
titleModuli space of polarized algebraic manifolds and Kähler metrics

journaltitleJournal of Geometry and Physics
titleTwo-dimensional gauge theories revisited
58D27 (14D20 58D29 81T13 81T40),paper

World Sci. Publ., River Edge, NJ

booktitleFields Medallists' Lectures
seriesWorld Sci. Ser. 20th Century Math.
titleRemarks on gauge theory, complex geometry and 4-manifold topology
57R57 (57N13 58D27 58G05)

American Mathematical Society

seriesMathematical Surveys and Monographs
titleThe Convenient Setting of Global Analysis
Infinite dimensional manifolds,paper

journaltitleCompositio Mathematica
titleFormules de localisation en cohomologie equivariante
53D20 (55N91 57S15),paper

groupsSymplectic connection
journaltitleSelecta Mathematica
titleContinuous cohomology of the group of volume-preserving and symplectic diffeomorphisms, measurable transfer and higher asymptotic cycles
Continuous cohomology,symplectomorphism groups.,paper

Elsevier BV

titleThe moment map and equivariant cohomology with generalized coefficients
53D20 (57S15 58A12),paper

Institut Mittag-Leffler

groupsSymplectic connection
issn0001-5962, 1871-2509
journaltitleActa Mathematica
titleUniqueness of Kähler-Ricci solitons

Tohoku University, Mathematical Institute

groupsSymplectic connection
issn0040-8735, 2186-585X
journaltitleTohoku Mathematical Journal
titleKähler-Einstein metrics for manifolds with nonvanishing Futaki character

booktitleSurv. Differ. Geom.
titleMoment maps in differential geometry

Univ. Paris 7

groupsSymplectic connection
titleSymplectomorphismes hamiltoniens et métriques kählériennes

journaltitleMonatsh. Math.
titleFlux homomorphisms and principal bundles over infinite dimensional manifolds
2000 Mathematics Subject Classification: 22E65; 58B20,53D50; Key words: Flux homomorphism,symplectic manifold,holonomy group,symplectomorphism,central extension,paper

groupsSymplectic connection
journaltitleActa Mathematica Hungarica
titleOn products of unbounded operators

Research Institute for Mathematical Sciences, Kyoto University

groupsSymplectic connection
journaltitleCommentarii Mathematici Helvetici
notearXiv:math/0112012 version: 6 type: article
titleCommutator length of symplectomorphisms
atics - Symplectic Geometry,Mathematics - Differential Geometry,Mathematics - Group Theory,Mathematics - Geometric Topology,53D22,53D05,53D40,53D45,paper

Birkhäuser Boston

titleMomentum Maps and Hamiltonian Reduction

International Press of Boston

groupsSymplectic connection
journaltitleComm. Anal. Geom.
titleMoment map, Futaki invariant and stability of projective manifolds
Hessian,momentum map squared,paper

* Entry 'Wang2004' (d:/Research/A_Central/Latex/bibtex/bib/Literature.bib): Invalid format 'CURRENT_TIMESTAMP' of date field 'urldate' - ignoring

Academia Press

booktitleLiber Amicorum Richard Delanghe: Een Veelzijdig Wiskundige
groupsSymplectic connection
titleMoment map for the space of symplectic connections

annotationComment: 33 pages. Corrections and a few paragraphs re-written. To appear in Jour. Sympl. Geom
journaltitleJournal of Symplectic Geometry
titleLocalization for the norm-square of the moment map and the two-dimensional Yang-Mills integral
omentum map squared,paper

annotationComment: 20 pages, to appear in Kodai Mathematical Journal
groupsSymplectic connection
journaltitleKodai Math. J.
notearXiv: math/0603706 version: 2
titleHarmonic total Chern forms and stability
atics - Differential Geometry,Mathematics - Symplectic Geometry,53C55,53C21,paper

groupsSymplectic connection
journaltitleLett. Math. Phys.
titleRemarks on symplectic connections
53D35,53D20,58J60,53C35,symplectic connection,Ricci-flat,symmetric symplectic spaces,Ricci-type connections,symplectic induction,paper

* Entry 'Gutt2006' (d:/Research/A_Central/Latex/bibtex/bib/Literature.bib): Invalid format '2019-02-21TZ' of date field 'urldate' - ignoring

Springer Japan

journaltitleJapan. J. Math.
titleTowards a Lie theory of locally convex groups

journaltitleJ. Lie Theory
titleCentral extensions of the Lie algebra of symplectic vector fields

groupsSymplectic connection
journaltitleAnn. Global Anal. Geom.
titleHessians of the Calabi functional and the norm function
Hessian,momentum map squared,paper

* Entry 'Wang2006' (d:/Research/A_Central/Latex/bibtex/bib/Literature.bib): Invalid format '2020-02-20TZ' of date field 'urldate' - ignoring

groupsSymplectic connection
journaltitleCommunications in Mathematical Physics
titleThe manifold of compatible almost complex structures and geometric quantization
Line Bundle,Toeplitz Operator,Symplectic Form,Parallel Transport,Geometric Quantization,paper

Publications CRM

subtitleAspects géométriques et algébriques, généralisations
titleL'algèbre et le groupe de Virasoro

Oxford University Press

groupsSymplectic connection
noteIncludes bibliographical references and index. - Formerly CIP
seriesOxford Math. Monogr.
titleSasakian Geometry

groupsSymplectic connection
journaltitleCommun. Math. Phys.
titleCanonical Sasakian metrics
Scalar Curvature,Einstein Metrics,Constant Scalar Curvature,Sasakian Manifold,Reeb Vector,paper

groupsSymplectic connection
journaltitleInt. J. Math.
titleExtremal almost-Kähler metrics
Mathematics - Differential Geometry,53B35,paper

editionSoftcover reprint of the original 2nd ed. 2004
groupsMomentum polytopes
titleTorus Actions on Symplectic Manifolds (Progress in Mathematics)

Basel: Springer

booktitleRecent trends in dynamical systems. Proceedings of the international conference, Munich, Germany, January 11–13, 2012, in honor of Jürgen Scheurle on the occasion of his 60th birthday
titleGradient flows in the normal and Kähler metrics and triple bracket generated metriplectic systems

journaltitleCommunications in Analysis and Geometry
titleLocalization for equivariant cohomology with varying polarization

seriesLecture Notes in Mathematics
titleDifferential Characters

De Gruyter Open Access

groupsSymplectic connection
journaltitleComplex Manifolds
titleOn the transverse Scalar Curvature of a Compact Sasaki Manifold
Transverse Scalar curvature,Symmetric space,Moment map,paper

European Mathematical Society - EMS - Publishing House GmbH

annotationComment: 40 pages; minor changes; submitted
groupsSymplectic connection
journaltitleCommentarii Mathematici Helvetici
notearXiv: 1105.5814
titleThe action homomorphism, quasimorphisms and moment maps on the space of compatible almost complex structures
Mathematics - Symplectic Geometry,Mathematics - Differential Geometry,Mathematics - Group Theory,paper

* Entry 'Shelukhin2014' (d:/Research/A_Central/Latex/bibtex/bib/Literature.bib): Invalid format '2020-03-12TZ' of date field 'urldate' - ignoring

American Mathematical Society (AMS)

journaltitleJournal of the American Mathematical Society
shorttitleKähler-Einstein metrics on Fano manifolds. I
titleKähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities

groupsSymplectic connection
journaltitleAnn. Global Anal. Geom.
titleInfinite dimensional moment map geometry and closed Fedosov’s star products

groupsSymplectic connection
journaltitleMathematische Zeitschrift
titleExtremal K-contact metrics
Moment map,K-Contact structure,Extremal metrics,Futaki-invariant,53D10,53B35,53D20,paper

annotationComment: 47 pages
journaltitleCommun. Pure Appl. Math.
notearXiv:1211.4669 [math] type: article
titleK-Stability and Kähler-Einstein metrics
Mathematics - Differential Geometry,Mathematics - Algebraic Geometry,Mathematics - Analysis of PDEs,53 Differential geometry,14 Algebraic geometry,paper

annotationComment: 15 pages; comments are welcome!
groupsSymplectic connection
notearXiv:1605.08019 [math] type: article
titleBoltzmann's entropy and Kähler-Ricci solitons
Mathematics - Differential Geometry,paper

International Press of Boston

groupsSymplectic connection
journaltitleAsian J. Math.
titleKähler–Ricci soliton and H-functional
Applied Mathematics,General Mathematics,paper

Springer International Publishing

booktitleGeometry, Analysis and Probability
groupsGIT, Symplectic connection
titleThe Ding functional, Berndtsson convexity and moment maps
Mathematics - Differential Geometry,53C55,paper

groupsSymplectic connection
titleCalabi’s Extremal Kähler Metrics: An Elementary Introduction
ive.org/web/20210108083652/https://cims.nyu.edu/ rodion/lib/P.

annotationComment: The curvature condition is added in the main result, and other modifications are made
groupsSymplectic connection
journaltitleJ. Symplectic Geom.
titleCahen-Gutt moment map, closed Fedosov star product and structure of the automorphism group
Mathematics - Differential Geometry,32Q15,46L65,53C55,paper

Cham: Springer

groupsGIT, Symplectic connection
isbn978-3-030-89299-9; 978-3-030-89300-2
seriesLect. Notes Math.
titleThe Moment-Weight Inequality and the Hilbert-Mumford Criterion. GIT from the Differential Geometric Viewpoint
.ethz.ch/ salamon/PREPRINTS/momentweight.pdf
Hessian,momentum map squared,paper

groupsSymplectic connection
journaltitlePure Appl. Math. Q.
titleOn the Kähler-Yang-Mills-Higgs equations

groupsSymplectic connection
journaltitleJournal of the European Mathematical Society
titleConformally Kähler, Einstein–Maxwell geometry

Universität Leipzig

annotationThe results of this thesis are contained in the the publications "Normal Form of Equivariant Maps in Infinite Dimensions", "Singular Symplectic Reduction In Infinite Dimension" (in preparation) and "Singular symplectic cotangent bundle reduction of gauge field theory".
groupsMy publications
titleNormal Form of Equivariant Maps and Singular Symplectic Reduction in Infinite Dimensions with Applications to Gauge Field Theory

groupsSymplectic connection
journaltitleJ. Symplectic Geom.
titleCritical symplectic connections on surfaces

groupsSymplectic connection
journaltitleMath. Z.
titleConformally Einstein-Maxwell Kähler metrics and structure of the automorphism group

groupsSymplectic connection
journaltitleInt. Math. Res. Notices
titleCoupled Kähler–Einstein metrics

groupsSymplectic connection
journaltitleJ. Symplectic Geom.
titleFutaki invariant for Fedosov's star products

groupsSymplectic connection
journaltitleJ. Geom. Anal.
titleAutomorphisms and deformations of conformally Kähler, Einstein-Maxwell metrics
Conformally Kähler,Einstein–Maxwell metric,Group of holomorphic automorphisms,Licherowicz operator,53C56,paper

groupsSymplectic connection
titleH-functional and Matsushima type decomposition theorem
Hessian,momentum map squared,paper

American Mathematical Society (AMS)

groupsSymplectic connection
journaltitleProc. Amer. Math. Soc.
titleHessian of the Ricci-Calabi functional
53C55 (53C25 58E11),paper

annotationComment: 21 pages. Corrected minor typos, included additional citations, clarified the definition of the coupled Futaki invariant
groupsSymplectic connection
journaltitleJ. Symplect. Geom.
notearXiv:1901.10454 [math] version: 2 type: article
titleOn coupled constant scalar curvature Kähler metrics

annotationThis is perhaps my best paper. We've submitted it to JAMS, where it was recently rejected. It is not directly used in the Research Project, but it shows that there are phenomena occurring in the infinite-dimensional setting: In finite dimensions, almost every action occurring in "nature" has a momentum map. In contrast, this work shows that classical momentum maps for diffeomorphism groups almost never exist.
groupsMy publications
titleGroup-valued momentum maps for actions of automorphism groups

Int. Press, Boston, MA

booktitleProceedings of the International Consortium of Chinese Mathematicians 2018
groupsSymplectic connection
titleDeformation quantization and Kähler geometry with moment map

journaltitleAsian J. Math.
titleComplex structures, moment maps, and the Ricci form

International Press of Boston

groupsSymplectic connection
journaltitleJournal of Symplectic Geometry
titleAlmost-Kähler smoothings of compact complex surfaces with $A_1$ singularities
53C55 (57R18),paper

annotationThis is the first paper in the series, where we construct Lie group extensions of diffeomorphism groups using the Kostant-Souriau prequantum construction. This time we use the space of embeddings $Emb(S, M)$. Not directly used in the Research Project.
groupsMy publications
journaltitleInt. Math. Res. Not.
titleCentral extensions of Lie groups preserving a differential form

Providence, RI: American Mathematical Society (AMS)

booktitleIntegrability, Quantization, and Geometry Ii. Quantum Theories and Algebraic Geometry. Dedicated to the Memory of Boris Dubrovin 1950–2019
isbn978-1-4704-5592-7; 978-1-4704-6435-6
titleA moment map interpretation of the Ricci form, Kähler–Einstein structures, and Teichmüller spaces

annotationComment: 19 pages
groupsSymplectic connection
titleMoment map, convex function and extremal point
Mathematics - Differential Geometry,Mathematics - Algebraic Geometry,Mathematics - Symplectic Geometry,paper

World Scientific Publishing Co.

groupsSymplectic connection
journaltitleInt. J. Math.
titleHermitian Calabi functional in complexified orbits
Hermitian Calabi functional,extremal almost Kähler metrics,complexified orbit,Hermitian Calabi flow,Hermitian Calabi functional,extremal almost Kähler metrics,complexified orbit,Hermitian Calabi flow,paper

groupsSymplectic connection
journaltitleAnnals of Global Analysis and Geometry
titleCalabi type functionals for coupled Kähler–Einstein metrics
Coupled Kähler–Einstein metric,Coupled Ding functional,Matsushima type decomposition theorem,Primary 53C25,Secondary 53C55,58E11,paper

groupsMy publications
pubstateIn preparation
titleSingular symplectic reduction in infinite dimensions

pubstateTo appear
titleInfinite-dimensional Lie Groups. General Theory and Main Examples


# The Sum of Squares function in the ring $\mathbb{Z}_{n}$

Rob Burns

###### Abstract

We consider the sum of squares function in the ring $\mathbb{Z}_{n}$. We
determine formulae in a number of cases when $n$ is a power of a prime.

##  1 Introduction

The representation of elements of a ring as a sum of squares (or more
generally as a sum of powers) is a very large and old subject. In the
following we attempt to give a flavour of the type of questions that have been
studied. It is not intended as a complete survey.

Restricting ourselves to the integers for the moment, Fermat proved that every
integer is either a square or a sum of $2$, $3$, or $4$ squares. Fermat also
discovered that an odd prime $p$ can be expressed as the sum of $2$ squares if
and only if $p\equiv 1\pmod{4}$. The first proof of this result was provided
by Euler using Fermat’s method of infinite descent ([12] [13] [15]). Euler
extended this result to show that a positive integer can be represented as the
sum of two squares if and only if each of its prime factors of the form $4k+3$
occurs as an even power ([35]).

Diophantus, Bachet, Fermat, Descartes and Lagrange amongst others worked on
the representation of integers as the sum of three or more squares. Legendre
proved in his Essai sur la theorie des nombres (1798) that a positive integer
can be written as a sum of three squares if and only if it is not of the form
$4^{x}(8y+7)$ for nonnegative integers $x$ and $y$.

In 1770, prior to Legendre’s result, Lagrange had proved that every positive
integer can be written as the sum of at most four squares. This is known as
Lagrange’s four-square theorem or Bachet’s conjecture ([20]).

The asymptotic density of the positive integers which are the sum of two
squares is zero. Landau showed that the asymptotic density of the positive
integers which are the sum of three squares is $\frac{5}{6}$ ([23]).

Moving away from the integers, we find that the representation of elements as
a sum of squares has been studied in many other rings.

In Hilbert’s book Foundations of Geometry (1899), he stated without proof that
a totally positive element of any number field can be expressed as a sum of
four squares in the field. Landau proved this for quadratic number fields in
1919 and Siegel extended the proof to all number fields in 1921 ([34]). The
situation is less straightforward when considering elements from the ring of
integers rather than general elements in the number field. For example, in the
field $\mathbb{Q}(i)$, which has $\mathbb{Z}(i)$ as its ring of integers, the
element $i$ cannot be expressed as a sum of squares of elements from
$\mathbb{Z}(i)$.

In 1940 Niven [29] investigated imaginary quadratic number fields
$K=\mathbb{Q}(\sqrt{-m})$ where $m$ is a square-free positive integer. If
$\mathbb{Z}_{K}$ is the ring of integers of $K$, Niven showed that every
element in $\mathbb{Z}_{K}$ can be expressed as a sum of three squares of
elements from $\mathbb{Z}_{K}$ if m$\equiv 3\pmod{4}$. He also showed that
when $m\equiv 1\pmod{4}$, an element $x+y\sqrt{-m}$ in $\mathbb{Z}_{K}$ is a
sum of three squares if and only if $y$ is an even integer.

Results for specific real quadratic number fields are known. For example,
Fritz Gotzky showed in [16] that every totally positive integer in
$K=\mathbb{Q}(\sqrt{5})$ can be represented as a sum of four squares of
integers in $K$. Harvey Cohn considered $K=\mathbb{Q}(\sqrt{2})$ in [8] and
proved that a totally positive integer $a=x+y\sqrt{2}$ can be written as a sum
of four squares of integers in $K$ if and only if $y$ is even. He in fact
provided a formula for the number of ways such an element $a$ could be written
as a sum of four integral squares in $K$. He also established a partial result
for $\mathbb{Q}(\sqrt{3})$.

In [7], Cogdell proved that, in a totally real number field, all sufficiently
large totally positive square free integers that are sums of three squares
locally everywhere are in fact sums of three squares globally. Schulze-Pillot
showed in his survey article [33] that the local-global principle does not
apply to all totally positive integers. In particular, if
$F=\mathbb{Q}(\sqrt{35})$ and $p$ is a prime satisfying
$\genfrac{(}{)}{}{}{p}{7}=1$, then no number of the form $7p^{2}$ is a sum of
three integral squares in F, even though $7$ is a sum of three integral
squares locally everywhere.

In [9], Colliot-Théléne and Xu consider the connection between the
representation of integral elements as a sum of squares and the Brauer-Manin
obstruction. An appendix to the paper provides an example of the local-global
principle in the setting of cyclotomic fields. The authors prove that an
element $x$ in the ring of integers $O$ of a cyclotomic field is a sum of
three squares of integers in $O$ if and only if $x$ is a sum of three squares
in all local completions $O_{v}$ of $O$.

Hilbert’s seventeenth problem asks whether every real positive definite
polynomial, i.e., one which takes positive values only, can be expressed as a
sum of squares of polynomials or, if not, whether it can be expressed as a sum
of squares of rational functions. There are examples of real positive definite
polynomials, such as the Motzkin polynomial
$(x^{2}+y^{2}-3z^{2})x^{2}y^{2}+z^{6}$, which cannot be expressed as a sum of
squares of polynomials. In 1927, Artin [2] showed that such polynomials can
always be expressed as a sum of squares of rational functions. Leep and Starr
[24] provided examples of positive semidefinite polynomials in two variables
which are sums of three rational squares, but not sums of polynomial squares.

The paper by Choi, Lam, Reznick and Rosenberg [6] discusses the connection
between representing an element $a$ in an integral domain $A$ as a sum of $n$
squares from the fraction field of $A$ and representing $a$ as a sum of $m\geq
n$ squares in $A$ itself. The authors prove that the former condition implies
the latter for certain unique factorisation domains, regular semilocal domains
and positive semidefinite polynomials in a polynomial ring over the reals.

The above questions can also be asked about matrix elements over a ring. In a
1968 paper, Carlitz [4] proved that every two-by-two matrix over $\mathbb{Z}$
is a sum of a three squares. Griffin and Krusemeyer [18] give some
circumstances under which a matrix is a sum of two squares. Newman [28]
treated the case of $n\times n$ matrices over $\mathbb{Z}_{2}$ and
$\mathbb{Z}$ and found the minimum number of squares needed to represent such
matrices. More recently Katre and Garge [22] examined $M_{n}(R)$, the $n\times
n$ matrices over $R$, where $R$ is a commutative, associative ring with unity.
They provided trace conditions on a matrix in $M_{n}(R)$ which ensure it is a
sum of $k$-th powers of matrices.

Hilbert’s seventeenth problem can be extended to matrices. Gondard and
Ribenboim [10] and Procesi and Schacher [31] independently proved that
matrices with positive semidefinite polynomial function entries can be
expressed as a sum of squares of symmetric matrices with rational function
entries.

In a general setting, Fernando, Ruiz and Scheiderer [14] showed that certain
excellent rings contain positive semidefinite elements which are not sums of
squares.

We finally turn to the ring $\mathbb{Z}_{n}$ of integers modulo $n$.
Harrington, Jones and Lamarche [19] determined the values of $n$ for which
every element of $\mathbb{Z}_{n}$ can be written as a sum of two squares. For
a prime $p$ and positive integer $k$, Burns [3] determined which elements of
$\mathbb{Z}_{p^{k}}$ can be written as a sum of two squares in the ring and
then again determined the values of $n$ for which all elements of
$\mathbb{Z}_{n}$ are a sum of two squares. In [1] Arias, Borja and Rubio
counted the number of integers in $\mathbb{Z}_{n}$ that are in the image of
various polynomials such as $x^{2}+y^{2}$ and $x^{2}+y^{2}+z^{2}$.

In the above contexts we also need to mention Waring’s Problem which asks for
the least positive integer $g$ such that every element of a given ring is the
sum of at most $g$ squares (or higher powers) of elements from the ring.
Waring’s problem has been studied on each of the rings mentioned above.

We now introduce the sum of squares function $r_{m}$ which counts the number
of ways an element of a ring can be written as a sum of $m$ squares, allowing
for zero as one or more of the squares. Specifically,

$r_{m}(n)=\lvert\\{(x_{1},x_{2},...,x_{m}):\sum_{i=1}^{m}x_{i}^{2}=n\\}\rvert.$

Much is known about $r_{m}$ over the integers. Jacobi expressed $r_{m}(n)$ in
terms of divisor functions when m = 2,4,6 and 8, For example, in [21] he
proved that

$r_{2}(n)=4(d_{1}(n)-d_{3}(n))$

where $d_{1}(n)$ and $d_{3}(n)$ are the number of divisors of $n$ congruent to
$1\pmod{4}$ and $3\pmod{4}$, respectively. Expressions for $r_{m}(n)$ are also
known for other values of $m$, however some of these involve terms that are
not explicitly given, as they appear only as coefficients of modular
functions. Milne [26] and Ono [30] established formulas for $r_{4s^{2}}$ and
$r_{4s^{2}+4s}$ for every $s\geq 1$. Asymptotic expressions are also known for
$r_{m}(n)$. The function $r_{2}$ is connected to the Gauss circle problem
through the summation function $R(N)=\sum_{n\leq N}r_{2}(n)$. The generating
function for $r_{m}$ can be expressed in terms of the Jacobi theta function:

$\sum_{n=0}^{\infty}r_{m}(n)x^{n}=\theta_{3}^{m}(x).$

This paper considers the sum of squares function over the ring
$\mathbb{Z}_{n}$ of integers modulo $n\in\mathbb{Z}$. We define the function
$N_{m}(t,n)$ to be the number of ways of expressing an element
$t\in\mathbb{Z}_{n}$ as a sum of $m$ squares of elements from
$\mathbb{Z}_{n}$. Let $\mathbb{Z}^{*}_{n}$ be the set of units in
$\mathbb{Z}_{n}$, i.e. the set of elements $t\in\mathbb{Z}^{*}_{n}$ such that
$\gcd(t,n)=1$. We then define the function $N^{*}_{m}(t,n)$ to be the number
of ways of expressing an element of $t\in\mathbb{Z}_{n}$ as a sum of $m$
squares of elements from $\mathbb{Z}^{*}_{n}$. We only need consider the case
$n=p^{k}$, where $p$ is prime and $k\geq 1$. The general result can then be
pieced together using the Chinese remainder theorem.

Lebesgue [11] calculated $N_{m}(t,p)$ when $p$ is an odd prime. His results
correspond to Theorems 1.5 \- 1.8 below. In 2014, Tóth [36] gave formulae for
$N_{2}(t,p)$, $N_{2}(t,p^{2})$ and $N_{m}(t,2^{k})$ where $p$ is prime and
$m,k\geq 1$. He also calculated $N_{m}(t,p^{k})$ for $t\in\\{0,1\\}$,
$m\in\\{2,3,4\\}$, $p$ prime and $k\geq 1$. Some of his results overlap with
Theorems 1.9 \- 1.12 below. Yang and Tang [37] gave a formula for $N_{2}(t,n)$
when $n$ is any integer and $t$ is any element of $\mathbb{Z}_{n}$. Their
results correspond to Theorems 1.3, 1.9 and 1.10 below. Calderón, Grau, Oller-
Marcén, and Tóth [5] calculated $N_{m}(t,p^{k})$ when $p$ is prime,
$\gcd(t,p)=1$ and $m,k\geq 1$. Grau and Oller-Marcén [17] completed the
picture by calculating $N_{m}(t,p^{k})$ for $p$ prime and $m,k\geq 1$.

Some authors have investigated $N^{*}_{m}(t,n)$. As far as we can tell, there
is no straightforward way of deriving one of $N^{*}_{m}(t,n)$ or $N_{m}(t,n)$
from the other. Yang and Tang [37] gave a formula for $N^{*}_{2}(t,n)$ when
$n$ is any integer and $t$ is any element of $\mathbb{Z}_{n}$. Mollahajiaghaei
[27] and Li and Ouyang [25] generalised this by establishing a formula for
$N^{*}_{m}(t,n)$ for any $m$. The formulae involve constants which are defined
recursively.

We start with results for the ring $\mathbb{Z}_{2^{k}}$.

######  Theorem 1.1.

Let $k,m\geq 1$. Then,

$N^{*}_{m}(t,2^{k})=\begin{cases}1&\text{if }\,\,k=1\,\,\text{ and
}\,\,t\equiv m\pmod{2}\\\ 2^{m}&\text{if }\,\,k=2\,\,\text{ and }\,\,t\equiv
m\pmod{4}\\\ 2^{2m+(m-1)(k-3)}&\text{if }\,\,k\geq 3\,\,\text{ and
}\,\,t\equiv m\pmod{8}\\\ 0&\text{otherwise.}\end{cases}$

######  Theorem 1.2.

$N_{m}(t,2)=2^{m-1}$ for $t\in\\{0,1\\}$. If $t\in\mathbb{Z}_{4}$, then,

$N_{m}(t,4)=\begin{cases}2^{2m-2}+(-1)^{t/2}2^{\frac{3m}{2}-1}\cos(\frac{m\pi}{4})&\text{if
}\,\,t\equiv 0\pmod{2}\\\
2^{2m-2}+(-1)^{(t-1)/2}2^{\frac{3m}{2}-1}\sin(\frac{m\pi}{4})&\text{if
}\,\,t\equiv 1\pmod{2}.\end{cases}$

######  Theorem 1.3.

Let $t\in\mathbb{Z}_{2^{k}}$ with $k\geq 2$. Write $t=2^{\alpha}\beta$ where
$2\nmid\beta$. Then in $\mathbb{Z}_{2^{k}}$,

$N_{2}(t,2^{k})=\begin{cases}2^{k}&\text{if }\,\,t=0\\\
2^{k+1}&\text{if}\,\,0\leq\alpha<k-1\text{ and }\beta\equiv 1\pmod{4}\\\
0&\text{if}\,\,0\leq\alpha<k-1\text{ and }\beta\equiv 3\pmod{4}\\\
2^{k}&\text{if}\,\,\alpha=k-1.\end{cases}$

######  Theorem 1.4.

Let $t\in\mathbb{Z}_{2^{k}}$ with $k\geq 3$ and write $t=2^{\alpha}\beta$
where $2\nmid\beta$. Then in $\mathbb{Z}_{2^{k}}$,

$N_{3}(t,2^{k})=\begin{cases}2^{3k/2}&\text{if }\,\,t=0\text{ and }k\text{ is
even }\\\ 2^{(3k+1)/2}&\text{if }\,\,t=0\text{ and }k\text{ is odd }\\\
3\times 2^{2k-\alpha/2-1}&\text{if}\,\,k-\alpha>2,\alpha\text{ is even and
}\beta\equiv 1\pmod{4}\\\ 2^{2k-\alpha/2}&\text{if}\,\,k-\alpha>2,\alpha\text{
is even and }\beta\equiv 3\pmod{8}\\\ 0&\text{if}\,\,k-\alpha>2,\alpha\text{
is even and }\beta\equiv 7\pmod{8}\\\ 3\times
2^{2k-(\alpha+1)/2}&\text{if}\,\,k-\alpha>2\text{ and }\alpha\text{ is odd
}\\\ 3\times 2^{3k/2}&\text{if}\,\,k-\alpha=2,\alpha\text{ is even and
}\beta\equiv 1\pmod{4}\\\ 2^{3k/2}&\text{if}\,\,k-\alpha=2,\alpha\text{ is
even and }\beta\equiv 3\pmod{4}\\\ 3\times
2^{(3k+1)/2}&\text{if}\,\,k-\alpha=2\text{ and }\alpha\text{ is odd }\\\
2^{(3k+1)/2}&\text{if}\,\,k-\alpha=1\text{ and }\alpha\text{ is even }\\\
3\times 2^{3k/2}&\text{if}\,\,k-\alpha=1\text{ and }\alpha\text{ is
odd.}\end{cases}$

Next we present results for the ring $\mathbb{Z}_{p}$, where $p$ is an odd
prime. When $t\in\mathbb{Z}_{p}$ the value of $N_{m}(t,p)$ depends on
$p\pmod{4}$, $m\pmod{2}$ and $\genfrac{(}{)}{}{}{t}{p}$. For clarity, we have
divided the result into four parts.

######  Theorem 1.5.

If $p$ is an odd prime with $p=1\pmod{4}$ and $m\geq 0$, then in
$\mathbb{Z}_{p}$,

$N_{2m+1}(t,p)=\begin{cases}p^{2m}&\text{if }\,\,t=0\\\
p^{2m}-p^{m}&\text{if}\,\,t\text{ is a non-residue}\pmod{p}\\\
p^{2m}+p^{m}&\text{if}\,\,t\text{ is a residue}\pmod{p}.\end{cases}$

######  Theorem 1.6.

If $p$ is an odd prime with $p=1\pmod{4}$ and $m\geq 1$, then in
$\mathbb{Z}_{p}$,

$N_{2m}(t,p)=\begin{cases}p^{2m-1}+p^{m}-p^{m-1}&\text{if }\,\,t=0\\\
p^{2m-1}-p^{m-1}&\text{if}\,\,t\neq 0.\end{cases}$

######  Theorem 1.7.

If $p$ is an odd prime with $p=3\pmod{4}$ and $m\geq 0$, then in
$\mathbb{Z}_{p}$

$N_{2m+1}(t,p)=\begin{cases}p^{2m}&\text{if }\,\,t=0\\\
p^{2m}+(-1)^{m+1}p^{m}&\text{if}\,\,t\text{ is a non-residue}\pmod{p}\\\
p^{2m}+(-1)^{m}p^{m}&\text{if}\,\,t\text{ is a residue}\pmod{p}.\end{cases}$

######  Theorem 1.8.

If $p$ is an odd prime with $p=3\pmod{4}$ and $m\geq 1$, then in
$\mathbb{Z}_{p}$

$N_{2m}(t,p)=\begin{cases}p^{2m-1}+(-1)^{m}p^{m}+(-1)^{m-1}p^{m-1}&\text{if
}\,\,t=0\\\ p^{2m-1}+(-1)^{m-1}p^{m-1}&\text{if}\,\,t\neq 0.\end{cases}$

If we restrict ourselves to the sum of two squares in the ring
$\mathbb{Z}_{p^{k}}$ we have the following:

######  Theorem 1.9.

Let $p$ be an odd prime with $p=1\pmod{4}$ and let $k\geq 1$. Let
$t\in\mathbb{Z}_{p^{k}}$ and write $t=p^{\alpha}\beta$ where $p\nmid\beta$.
Then in $\mathbb{Z}_{p^{k}}$

$N_{2}(t,p^{k})=\begin{cases}p^{k-1}\left(p(k+1)-k\right)&\text{if }\,\,t=0\\\
(\alpha+1)(p-1)p^{k-1}&\text{if}\,\,t\neq 0\end{cases}$

######  Theorem 1.10.

Let $p$ be an odd prime with $p=3\pmod{4}$ and let $k\geq 1$. Let
$t\in\mathbb{Z}_{p^{k}}$ and write $t=p^{\alpha}\beta$ where $p\nmid\beta$.
Then in $\mathbb{Z}_{p^{k}}$

$N_{2}(t,p^{k})=\begin{cases}p^{2\lfloor\frac{k}{2}\rfloor}&\text{if
}\,\,t=0\\\ (p+1)p^{k-1}&\text{if}\,\,\alpha\,\,\text{is even}\\\
0&\text{if}\,\,\alpha\,\,\text{is odd}\end{cases}$

For the sum of three squares in the ring $\mathbb{Z}_{p^{k}}$ we have:

######  Theorem 1.11.

Let $p$ be an odd prime with $p=1\pmod{4}$ and let $k\geq 1$. Let
$t\in\mathbb{Z}_{p^{k}}$ and write $t=p^{\alpha}\beta$ where $p\nmid\beta$.
Then in $\mathbb{Z}_{p^{k}}$

$N_{3}(t,p^{k})=\begin{cases}p^{2k}+p^{2k-1}-p^{\lceil\frac{3k}{2}\rceil-1}&\text{if
}\,\,t=0\\\
(p^{2k-1}-p^{2k-\frac{\alpha+3}{2}})(p+1)&\text{if}\,\,\alpha\text{ is odd}\\\
p^{2k-1}(p+1)&\text{if}\,\,\alpha\,\,\text{ is even
and}\,\,\genfrac{(}{)}{}{}{\beta}{p}=1\\\
p^{2k-1}(p+1)-2p^{2k-1-\frac{\alpha}{2}}&\text{if}\,\,\alpha\,\,\text{ is even
and}\,\,\genfrac{(}{)}{}{}{\beta}{p}=-1.\end{cases}$

######  Theorem 1.12.

Let $p$ be an odd prime with $p=3\pmod{4}$ and let $k\geq 1$. Let
$t\in\mathbb{Z}_{p^{k}}$ and write $t=p^{\alpha}\beta$ where $p\nmid\beta$.
Then in $\mathbb{Z}_{p^{k}}$,

$N_{3}(t,p^{k})=\begin{cases}p^{2k}+p^{2k-1}-p^{\lceil\frac{3k}{2}\rceil-1}&\text{if
}\,\,t=0\\\
(p^{2k-1}-p^{2k-\frac{\alpha+3}{2}})(p+1)&\text{if}\,\,\alpha\,\,\text{ is
odd}\\\ p^{2k-1}(p+1)-2p^{2k-1-\frac{\alpha}{2}}&\text{if}\,\,\alpha\,\,\text{
is even and}\,\,\genfrac{(}{)}{}{}{\beta}{p}=1\\\
p^{2k-1}(p+1)&\text{if}\,\,\alpha\,\,\text{ is even
and}\,\,\genfrac{(}{)}{}{}{\beta}{p}=-1.\end{cases}$

##  2 Notation

We will make use of the Legendre symbol $\genfrac{(}{)}{}{}{t}{p}$. The floor
of the real number $x$ is written $\lfloor x\rfloor$ and defined as the
largest integer $\leq x$. The ceiling of the real number $x$ is written
$\lceil x\rceil$ and defined as the smallest integer $\geq x$. If $A$ is a
set, we denote the number of elements in $A$ by $\lvert A\rvert$.

When discussing divisibility of an integer $x$ by a prime $p$, we will use the
notation $p^{n}\|x$ to mean that $n$ is the highest power of $p$ dividing $x$,
i.e. $p^{n}\mid x$ and $p^{n+1}\nmid x$. We will also use the notation
$\operatorname{ord}_{p}(x)=n$ to mean that $n$ is the highest power of $p$
dividing $x$.

For integers $n\in\mathbb{N}$, we define the sets $S_{m}(t,n)$ and
$S^{*}_{m}(t,n)$ as follows:

$S_{m}(t,n):=\,\,\Big{\\{}(x_{1},x_{2},...,x_{m})\in\mathbb{Z}_{n}:\sum_{i=1}^{m}x_{i}^{2}\equiv
t\pmod{n}\Big{\\}}$ (1)

$S^{*}_{m}(t,n):=\,\,\Big{\\{}(x_{1},x_{2},...,x_{m})\in\mathbb{Z}^{*}_{n}:\sum_{i=1}^{m}x_{i}^{2}\equiv
t\pmod{n}\Big{\\}}$ (2)

We then have $N_{m}(t,n)=\lvert S_{m}(t,n)\rvert$ and $N^{*}_{m}(t,n)=\lvert
S^{*}_{m}(t,n)\rvert$.

Our first observation is that the value of $N_{m}(t,p)$ depends only on
$\genfrac{(}{)}{}{}{t}{p}$.

######  Lemma 2.1.

Let $m\geq 1$ and suppose $s,t\in\mathbb{Z}_{p}$ with
$\genfrac{(}{)}{}{}{s}{p}=\genfrac{(}{)}{}{}{t}{p}$. Then
$N_{m}(s,p)=N_{m}(t,p)$.

###### Proof.

If $\genfrac{(}{)}{}{}{s}{p}=\genfrac{(}{)}{}{}{t}{p}$ then $t/s$ is a
quadratic residue $\pmod{p}$. Let $z=\sqrt{t/s}$ in $\mathbb{Z}_{p}$. Then

$\sum_{i=1}^{m}(x_{i})^{2}=s\iff\sum_{i=1}^{m}(x_{i}*z)^{2}=t.$

∎

According to the lemma above, all information in $\mathbb{Z}_{p}$ about the
function $N_{m}$ is contained in a $3\times 1$ vector which we will call
$\gamma_{m}$ and define by

$\gamma_{m}:=\begin{pmatrix}N_{m}(0,p)\\\ N_{m}(s,p)\\\
N_{m}(t,p)\end{pmatrix}$ (3)

where $s$ is any non-residue and $t$ is any residue $\pmod{p}$. We denote the
components of $\gamma_{m}$ by $\gamma_{m,1}$, $\gamma_{m,2}$ and
$\gamma_{m,3}$.

##  3 The $\mathbb{Z}_{2^{k}}$ case: proof of theorems 1.1 \- 1.4

###  3.1 Preliminaries

We first provide some results which are required for the proof of theorems 1.1
\- 1.4. The following classical result was published in 1834 [32].

$\sum_{j}\binom{k}{t+nj}=\frac{1}{n}\sum_{j=1}^{n}(2\cos(\frac{j\pi}{n}))^{k}\cos(\frac{(k-2t)j\pi}{n})$
(Ramus’ Identity)

######  Lemma 3.1.

Let $t\in\mathbb{Z}_{2^{k}}$ with $k\geq 3$ and $2\nmid t$. Then,

$N_{1}(t,2^{k})=\Big{\lvert}\Big{\\{}x\in\mathbb{Z}_{2^{k}}:x^{2}\equiv
t\pmod{2^{k}}\Big{\\}}\Big{\rvert}=\begin{cases}4&\text{if }\,\,t\equiv
1\pmod{8}\\\ 0&\text{otherwise.}\end{cases}$

###### Proof.

It is clear that any element which is $\,\not\equiv 1\pmod{8}$ cannot be a
square in $\mathbb{Z}_{2^{k}}$. We will use induction on $k$ to show that any
element of $\mathbb{Z}_{2^{k}}$ which is $\equiv 1\pmod{8}$ is a square in
$\mathbb{Z}_{2^{k}}$. This holds for $k=3$. Suppose it holds for $k$ and let
$t\in\mathbb{Z}_{2^{k+1}}$ with $t\equiv 1\pmod{8}$. By induction there is a
$y$ such that $y^{2}\equiv t\pmod{2^{k}}$. So $t=y^{2}+2^{k}r$ for some $r$.
If $r$ is even then $y^{2}\equiv t\pmod{2^{k+1}}$. If $r$ is odd, then

$\displaystyle(y+2^{k-1})^{2}=$ $\displaystyle\,\,y^{2}+2^{k}y+2^{2k-2}$
$\displaystyle=$ $\displaystyle\,\,t+2^{k}(y-r+2^{k-2}).$

Since both $y$ and $r$ are odd, the term in brackets is even and so
$(y+2^{k-1})^{2}\equiv t\pmod{2^{k+1}}$. This completes the induction.

Next, we show that, for each $t\in\mathbb{Z}_{2^{k}}$ with $t\equiv
1\pmod{8}$, there are at least 4 elements $y\in\mathbb{Z}_{2^{k}}$ with
$y^{2}\equiv t\pmod{2^{k}}$. From above, we know there is at least one such
$y$ value. The elements $2^{k}-y$, $2^{k-1}\pm y$ are all distinct in
$\mathbb{Z}_{2^{k}}$ and their squares are $\equiv t\pmod{2^{k}}$. Therefore,

$\Big{\lvert}\Big{\\{}x\in\mathbb{Z}_{2^{k}}:x^{2}\equiv
t\pmod{2^{k}}\Big{\\}}\Big{\rvert}\geq 4.$

Since there are $2^{k-1}$ odd elements in $\mathbb{Z}_{2^{k}}$ and $2^{k-3}$
elements which are $\equiv 1\pmod{8}$, there can be no values of $t$ with

$\Big{\lvert}\Big{\\{}x\in\mathbb{Z}_{2^{k}}:x^{2}\equiv
t\pmod{2^{k}}\Big{\\}}\Big{\rvert}>4.$

The lemma follows. ∎

######  Lemma 3.2.

$N_{2}(0,2^{k})=2^{k}$ when $k\geq 1$.

###### Proof.

For any $x,y\in\mathbb{Z}_{2^{k}}$,

$\operatorname{ord}_{2}(x^{2}+y^{2})=\begin{cases}\min\\{\operatorname{ord}_{2}(x^{2}),\operatorname{ord}_{2}(y^{2})\\}&\,\,\text{if
}\,\,\operatorname{ord}_{2}(x^{2})\neq\operatorname{ord}_{2}(y^{2})\\\
1+\min\\{\operatorname{ord}_{2}(x^{2}),\operatorname{ord}_{2}(y^{2})\\}&\,\,\text{if
}\,\,\operatorname{ord}_{2}(x^{2})=\operatorname{ord}_{2}(y^{2}).\end{cases}$

Suppose $(x,y)\in S_{2}(0,2^{k})$.

If $k$ is even, then either $x=0$ or $\operatorname{ord}_{2}(x)\geq k/2$. The
same applies to $y$. Now,

$\\{x:\operatorname{ord}_{2}(x)\geq
k/2\\}=\\{2^{k/2}j:j=1,2,\dots,2^{k-k/2}-1\\}.$

Counting $x=0$, there are therefore $2^{k/2}$ choices for $x$ and the same
number for $y$. We then have $N_{2}(0,2^{k})=2^{k/2}\times 2^{k/2}=2^{k}$.

If $k$ is odd, then either
$x,y\in\\{2^{(k+1)/2}j:j=0,1,2,\dots,2^{k-(k+1)/2}-1\\}$ or
$\operatorname{ord}_{2}(x)=\operatorname{ord}_{2}(y)=(k-1)/2$. We have,

$\operatorname{ord}_{2}(x)=(k-1)/2\iff
x\in\\{2^{(k-1)/2}j:j=1,3,5,2^{(k+1)/2}-1\\}.$

So, when $k$ is odd,

$N_{2}(0,2^{k})=2^{(k-1)/2}\times 2^{(k-1)/2}+2^{(k-1)/2}\times
2^{(k-1)/2}=2^{k}.$

∎

######  Lemma 3.3.

When $k\geq 1$, we have

$N_{3}(0,2^{k})=\begin{cases}2^{3k/2}&\text{if }\,\,k\text{ is even}\\\
2^{(3k+1)/2}&\text{if }\,\,k\text{ is odd}.\\\ \end{cases}$

###### Proof.

The result when $k<3$ is clear so assume $k\geq 3$.

$\lvert\\{(x,y,z):x^{2}+y^{2}+z^{2}\equiv
0\pmod{2^{k}}\\}\rvert=\sum_{z=0}^{2^{k}-1}\lvert\\{(x,y):x^{2}+y^{2}\equiv-z^{2}\pmod{2^{k}}\\}\rvert.$

We break this sum into two parts. The first part contains values of $z$ for
which $z^{2}\equiv 0\pmod{2^{k}}$. There are $2^{\lfloor k/2\rfloor}$ values
of $z$ such that $z^{2}\equiv 0\pmod{2^{k}}$. From lemma 3.2
$\lvert\\{(x,y):x^{2}+y^{2}\equiv 0\pmod{2^{k}}\\}\rvert=2^{k}$. The first
part of the sum is therefore equal to $2^{k+\lfloor k/2\rfloor}$.

The second part of the sum contains non-zero values of $z^{2}$. For such $z$
we can write $-z^{2}=2^{2r}s\pmod{2^{k}}$, where $2r<k$ and $s\equiv
7\pmod{8}$. For this value of $z$, we have from theorem 1.3

$N_{2}(-z^{2},2^{k})=\begin{cases}2^{k}&\,\,\text{if}\,\,2r=k-1\\\
0&\,\,\text{ otherwise }.\end{cases}$

If $k$ is even, no such $z$ values exist. If k is odd,

$\displaystyle\lvert\\{$ $\displaystyle
z:-z^{2}=2^{k-1}s\pmod{2^{k}}\,\,\text{ with }s\equiv 7\pmod{8}\\}\rvert$
$\displaystyle=\,\,$ $\displaystyle\lvert\\{z:z=2^{(k-1)/2}u\,\,\text{ with
}\,\,u\,\,\text{ odd and }\,\,1\leq u<2^{(k+1)/2}\\}\rvert$
$\displaystyle=\,\,$ $\displaystyle 2^{(k-1)/2}.$

The second part of the sum is therefore equal to zero when $k$ is even and
$2^{k}\times 2^{(k-1)/2}=2^{(3k-1)/2}$ when $k$ is odd. Combining the two
parts of the sum produces the result. ∎

######  Lemma 3.4.

Let $t\in\mathbb{Z}_{2^{k}}$ with $k\geq 1$ and $2\nmid t$. Then

$N_{2}(t,2^{k})=\begin{cases}2&\text{if }\,\,k=1\\\ 2^{k+1}&\text{if
}\,\,t\equiv 1\pmod{4}\,\,\text{ and }\,\,k\geq 2\\\ 0&\text{if }\,\,t\equiv
3\pmod{4}\,\,\text{ and }\,\,k\geq 2.\end{cases}$

###### Proof.

We assume $k\geq 3$ since it is easy to see that $N_{2}(1,2)=2$,
$N_{2}(1,4)=8$ and $N_{2}(3,4)=0$. If $t\in\\{3,7\\}\mod 8$, then there are no
$x,y$ such that $x^{2}+y^{2}\equiv t\pmod{8}$ and a fortiori
$N_{2}(t,2^{k})=0$.

Suppose $t\equiv 1\pmod{8}$. If $(x,y)\in S_{2}(t,2^{k})$, then either
$x^{2}\equiv 1\pmod{8}$ and $y^{2}\equiv 0\pmod{8}$ or vice versa. Now,

$\lvert\\{y\in\mathbb{Z}_{2^{k}}:y^{2}\equiv
0\pmod{8}\\}\rvert=\lvert\\{y\in\mathbb{Z}_{2^{k}}:y\equiv
0\pmod{4}\\}\rvert=2^{k-2}.$

For each such $y$, $t-y^{2}\equiv 1\pmod{8}$. By lemma 3.1 there are $4$
values of $x\in\mathbb{Z}_{2^{k}}$ satisfying the congruence $x^{2}\equiv
t-y^{2}\pmod{2^{k}}$. Hence, when $t\equiv 1\pmod{8}$,

$N_{2}(t,2^{k})=4\times 2^{k-2}+2^{k-2}\times 4=2^{k+1}.$

Suppose finally that $t\equiv 5\pmod{8}$. If $(x,y)\in S_{2}(t,2^{k})$, then
either $x^{2}\equiv 1\pmod{8}$ and $y^{2}\equiv 4\pmod{8}$ or vice versa. The
same argument as above can be used again.∎

######  Lemma 3.5.

Let $t\in\mathbb{Z}_{2^{k}}$ with $k\geq 2$ and $2\nmid t$. Then

$N_{2}(2t,2^{k})=\begin{cases}4&\,\,\text{if }\,\,k=2\\\ 2^{k+1}&\,\,\text{if
}\,\,t\equiv 1\pmod{4}\,\,\text{and }k\geq 3\\\ 0&\,\,\text{if }\,\,t\equiv
3\pmod{4}\,\,\text{and }k\geq 3.\end{cases}$

###### Proof.

We assume $k\geq 3$ since the result for $k=2$ is straightforward. If $t\equiv
3\pmod{4}$ then $2t\equiv 6\pmod{8}$ and $N_{2}(2t,2^{k})=0$.

Suppose $t\equiv 1\pmod{4}$. Then $2t\equiv 2\pmod{8}$. If $(x,y)\in
S_{2}(t,2^{k})$, then both $x^{2}$ and $y^{2}$ must be $\equiv 1\pmod{8}$.
Now,

$\lvert\\{y\in\mathbb{Z}_{2^{k}}:y^{2}\equiv
1\pmod{8}\\}\rvert=\lvert\\{y\in\mathbb{Z}_{2^{k}}:y\equiv
1\pmod{2}\\}\rvert=2^{k-1}.$

For each such $y$, $t-y^{2}\equiv 1\pmod{8}$. By lemma 3.1 there are $4$
values of $x\in\mathbb{Z}_{2^{k}}$ satisfying the congruence $x^{2}\equiv
t-y^{2}\pmod{2^{k}}$. Hence, when $t\equiv 1\pmod{4}$,

$N_{2}(2t,2^{k})=4\times 2^{k-1}=2^{k+1}.$

∎

######  Lemma 3.6.

Let $t\in\mathbb{Z}_{2^{k}}$ with $k\geq 2$ and $2\nmid t$. Then

$N_{3}(t,2^{k})=\begin{cases}3\times 2^{2k-1}&\text{if }\,\,t\equiv
1\pmod{4}\\\ 8&\text{if }\,\,t\equiv 3\pmod{4}\,\,\text{ and }\,\,k=2\\\
2^{2k}&\text{if }\,\,t\equiv 3\pmod{8}\,\,\text{ and }\,\,k\geq 3\\\
0&\text{if }\,\,t\equiv 7\pmod{8}.\end{cases}$

###### Proof.

The result is clear when $k=2$ and when $t\equiv 7\pmod{8}$. Assume $k\geq 3$.
If $x^{2}+y^{2}+z^{2}\equiv t\pmod{2^{k}}$, then either all of $\\{x,y,z\\}$
are odd, or two of $\\{x,y,z\\}$ are even and the other odd. We will treat
each possibility separately. We first count the number of suitable
$\\{x,y,z\\}$ such that all are odd. In this case, we must have $t\equiv
3\pmod{8}$. The number of such solutions is:

$N_{3}(t,2^{k})=\sum_{z\,\,\text{ odd
}}\lvert\\{(x,y)\in\mathbb{Z}_{2^{k}}:x,y\,\,\text{ odd }\,\,\text{ and
}\,\,x^{2}+y^{2}\equiv t-z^{2}\pmod{2^{k}}\\}\rvert$

Since $t-z^{2}\equiv 2\pmod{8}$ for odd $z$, we can write $z=2u+1$ and use
lemma 3.5 to show that:

$N_{3}(t,2^{k})=2^{k+1}\times\lvert\\{u:0\leq u<2^{k-1}\\}\rvert=2^{2k}.$

Secondly, we count the number of solutions in which exactly two of
$\\{x,y,z\\}$ are even. In this case, $t\equiv 1\pmod{4}$. The element $z$ is
even in two out of the three possible arrangements of odd/even elements.
Writing $z=2u$, and using lemma 3.4, the number of such solutions when
$t\equiv 1\pmod{4}$ is

$\displaystyle N_{3}(t,2^{k})$
$\displaystyle=\frac{3}{2}\times\sum_{u=0}^{2^{k-1}-1}\lvert\\{(x,y)\in\mathbb{Z}_{2^{k}}:\,\,x^{2}+y^{2}\equiv
t-4u^{2}\pmod{2^{k}}\\}\rvert$ $\displaystyle=\frac{3}{2}\times 2^{k+1}\times
2^{k-1}$ $\displaystyle=3\times 2^{2k-1}.$

∎

######  Lemma 3.7.

Let $t\in\mathbb{Z}_{2^{k}}$ with $k\geq 2$ and $2\nmid t$. Then
$N_{3}(2t,2^{k})=3\times 2^{2k-1}$.

###### Proof.

If $x^{2}+y^{2}+z^{2}\equiv 2t\pmod{2^{k}}$, then two of $\\{x,y,z\\}$ must be
odd and the other even. So,

$\displaystyle N_{3}(2t,2^{k})\,\,=$
$\displaystyle\,\,\lvert\\{(x,y,z)\in\mathbb{Z}_{2^{k}}:x^{2}+y^{2}+z^{2}\equiv
2t\pmod{2^{k}}\\}\rvert$ $\displaystyle=$
$\displaystyle\,\,3\times\sum_{u=0}^{2^{k-1}-1}\lvert\\{(x,y)\in\mathbb{Z}_{2^{k}}:x^{2}+y^{2}\equiv
2(t-2u^{2})\pmod{2^{k}}\\}\rvert.$

When $u$ is even, $t-2u^{2}\equiv t\pmod{4}$ and when $u$ is odd,
$t-2u^{2}\equiv t-2\pmod{4}$. By lemma 3.5, we have,

$\displaystyle N_{3}(2t,2^{k})$
$\displaystyle=3\times\sum_{u\,\,\text{even}}\lvert\\{(x,y)\in\mathbb{Z}_{2^{k}}:x^{2}+y^{2}\equiv
2t\pmod{2^{k}}\\}\rvert$
$\displaystyle+\,\,3\times\sum_{u\,\,\text{odd}}\lvert\\{(x,y)\in\mathbb{Z}_{2^{k}}:x^{2}+y^{2}\equiv
2(t-2)\pmod{2^{k}}\\}\rvert$ $\displaystyle=3\times 2^{k-2}\times 2^{k+1}$
$\displaystyle=3\times 2^{2k-1}.$

∎

######  Lemma 3.8.

Let $t\in\mathbb{Z}_{2^{k}}$ with $k\geq 3$ and $t=2^{r}s$ with $2\nmid s$. If
$r\geq 2$, then $N_{2}(t,2^{k})=4\times N_{2}(2^{r-2}s,2^{k-2})$ and
$N_{3}(t,2^{k})=8\times N_{3}(2^{r-2}s,2^{k-2})$.

###### Proof.

We introduce the map $f:N_{2}(2^{r}s,2^{k})\to N_{2}(2^{r-2}s,2^{k-2})$ given
by $f((x,y))=(x/2\pmod{2^{r-2}},y/2\pmod{2^{r-2}})$. The map is well defined
because, if $(x,y)\in S_{2}(2^{r}s,2^{k})$ for $k\geq 3$ and $r\geq 2$, then
both $x$ and $y$ must be even. The map is onto since, if $(x,y)\in
S_{2}(2^{r-2}s,2^{k-2})$, then $(2x,2y)\in S_{2}(2^{r}s,2^{k})$ and
$f((2x,2y))=(x,y)$. The map is also $4$ to $1$. If $(x,y)\in
S_{2}(2^{r}s,2^{k})$, then so are $(x+2^{r-1},y),(x,y+2^{r-1})$ and
$(x+2^{r-1},y+2^{r-1})$ and all four elements are mapped to the same point by
$f$. If $f((x,y))=f((u,v))$ then $u/2\equiv x/2\pmod{2^{r-2}}$ so $u\equiv
x\pmod{2^{r-1}}$. Therefore, $u\pmod{2^{r}}\in\\{x,x+2^{r-1}\\}$. Similarly,
$v\pmod{2^{r}}\in\\{y,y+2^{r-1}\\}$. Therefore, $(u,v)$ must be one of the
four elements $(x,y),(x+2^{r-1},y),(x,y+2^{r-1})$ or $(x+2^{r-1},y+2{r-1})$.
The lemma follows.

With suitable changes, the same argument works for $N_{3}(t,2^{k})$. ∎

###  3.2  $N^{*}_{m}(t,2^{k})$ : Proof of theorem 1.1

We note that theorem 1.1 appears as Theorem 6.2 in [27] and Theorem 4.4 in
[25].

The proof when $k\in\\{1,2\\}$ is obvious so assume $k\geq 3$. When $x$ is
odd, $x^{2}\equiv 1\pmod{8}$. Therefore, there are no solutions when
$t\not\equiv m\pmod{8}$.

The proof is by induction on $m$. The theorem holds for $m=1$ by Lemma 3.1.
Suppose the theorem holds for $m-1$. Then, if $t\equiv m\pmod{8}$,

$\displaystyle N^{*}_{m}(t,2^{k})=$
$\displaystyle\,\,\Big{\lvert}\Big{\\{}(x_{1},x_{2},...,x_{m})\in\mathbb{Z}^{*}_{2^{k}}:\sum_{i=1}^{m}x_{i}^{2}\equiv
t\pmod{2^{k}}\Big{\\}}\Big{\rvert}$ $\displaystyle=$
$\displaystyle\,\,\sum_{s}N^{*}_{m-1}(s,2^{k})\times N^{*}_{1}(t-s,2^{k})$
$\displaystyle=$ $\displaystyle\,\,4\times\sum_{s\equiv
m-1\pmod{8}}N^{*}_{m-1}(s,2^{k})\,\,\text{ by Lemma \ref{x2=1mod8}}$
$\displaystyle=$ $\displaystyle\,\,4\times 2^{k-3}\times
2^{2(m-1)+(m-2)(k-3)}\,\,\text{ by the inductive step}$ $\displaystyle=$
$\displaystyle\,\,2^{2m+(m-1)(k-3)}.$

This finishes the proof.

###  3.3  $N_{m}(t,2)$ and $N_{m}(t,4)$ : Proof of theorem 1.2

Firstly, if $t,x_{i}\in\mathbb{Z}_{2}$ for $i=1,2,\dots,m$, then

$\displaystyle\sum_{i=1}^{m}x_{i}^{2}\equiv t\pmod{2}\iff$
$\displaystyle\sum_{i=1}^{m}x_{i}\equiv t\pmod{2}$ $\displaystyle\iff$
$\displaystyle\lvert\\{x_{i}:x_{i}=1\\}\rvert\equiv t\pmod{2}.$

So,

$N_{m}(t,2)=\sum_{j}\binom{m}{t+2j}.$

The result for $N_{m}(t,2)$ follows from the following identities, which are
special cases of (Ramus’ Identity):

$\sum_{j}\binom{m}{2j}=\sum_{j}\binom{m}{1+2j}=2^{m-1}.$

Similarly, if $t,x_{i}\in\mathbb{Z}_{4}$ for $i=1,2,\dots,m$, then

$\sum_{i=1}^{m}x_{i}^{2}\equiv t\pmod{4}\iff\lvert\\{x_{i}:x_{i}\equiv
1\pmod{2}\\}\rvert\equiv t\pmod{4}$

So,

$N_{m}(t,4)\,\,=\,\,\sum_{j}\binom{m}{t+4j}2^{t+4j}\,\,2^{m-t-4j}\,\,=\,\,2^{m}\sum_{j}\binom{m}{t+4j}.$

The result for $N_{m}(t,4)$ follows from another application of (Ramus’
Identity).

###  3.4  $N_{2}(t,2^{k})$ : Proof of theorem 1.3

The result is clear when $k=2$ so assume $k\geq 3$.

The $t=0$ case is covered by lemma 3.2. For non-zero $t$, write
$t=2^{\alpha}\beta$ where $2\nmid\beta$. We use proof by descent. By lemma
3.8, $N_{2}(t,2^{k})=4\times N_{2}(2^{\alpha-2}\beta,2^{k-2})$ when
$\alpha\geq 2$. Continuing the reduction, we get

$N_{2}(t,2^{k})=\begin{cases}4^{\alpha/2}\times
N_{2}(\beta,2^{k-\alpha})&\text{if }\,\,\alpha\equiv 0\pmod{2}\\\
4^{(\alpha-1)/2}\times N_{2}(2\beta,2^{k-\alpha+1})&\text{if }\,\,\alpha\equiv
1\pmod{2}.\end{cases}$

We can then use lemmas 3.4 and 3.5 to derive the expressions for
$N_{2}(t,2^{k})$ given in the theorem.

###  3.5  $N_{3}(t,2^{k})$ : Proof of theorem 1.4

The proof of theorem 1.4 follows the same pattern as the proof of theorem 1.3.
The result for $t=0$ is contained in lemma 3.3. For non-zero $t$, write
$t=2^{\alpha}\beta$ where $2\nmid\beta$. We again use proof by descent. By
lemma 3.8, $N_{3}(t,2^{k})=8\times N_{3}(2^{\alpha-2}\beta,2^{k-2})$ when
$\alpha\geq 2$. Continuing the reduction, we get

$N_{3}(t,2^{k})=\begin{cases}8^{\alpha/2}\times
N_{3}(\beta,2^{k-\alpha})&\text{if }\,\,\alpha\equiv 0\pmod{2}\\\
8^{(\alpha-1)/2}\times N_{3}(2\beta,2^{k-\alpha+1})&\text{if }\,\,\alpha\equiv
1\pmod{2}.\end{cases}$

We can then use lemmas 3.6 and 3.7 to derive the expressions for
$N_{3}(t,2^{k})$ given in the theorem.

##  4 The $\mathbb{Z}_{p}$ case: proof of theorems 1.5 \- 1.8

In this section we will prove the four theorems 1.5, 1.6, 1.7 and 1.8 related
to $\mathbb{Z}_{p}$. We first prove the theorems for $N_{1}$ and $N_{2}$ and
then establish a recurrence relation which can be used to derive the general
formulae. Using the definition of $\gamma$ given in (3), it is easy to see
that

$\gamma_{1}=\begin{pmatrix}1\\\ 0\\\ 2\end{pmatrix}$ (4)

which establishes theorems 1.5 and 1.7 for $N_{1}$.

We include the proof of the case $m=2$ here even though it is available
elsewhere. It requires a few lemmata.

######  Lemma 4.1.

For each non-zero $t\in\mathbb{Z}_{p}$ the congruence $y^{2}=x^{2}+t\pmod{p}$
has $p-1$ solutions $(x,y)$.

###### Proof.

Rearranging we have $y^{2}-x^{2}=t\pmod{p}$. Factorising the LHS and changing
variables to $u=y-x$, $v=y+x$, which is an invertible map in $\mathbb{Z}_{p}$,
we have $uv=t\pmod{p}$. For each non-zero choice for $\,\,u\pmod{p}$ there is
a unique value for $v=t*u^{-1}\pmod{p}$ which satisfies the congruence. There
are therefore $p-1$ solutions for $(u,v)$ and the number of solutions of the
original congruence in terms of the variables $(x,y)$ is the same as the
number of solutions for $(u,v)$. ∎

######  Lemma 4.2.

For fixed non-zero $t\in\mathbb{Z}_{p}$,
$\sum_{x=0}^{p-1}\genfrac{(}{)}{}{}{x^{2}+t}{p}=-1$.

###### Proof.

On the one hand the lemma 4.1 says the number of solutions to the congruence
$y^{2}=x^{2}+t\pmod{p}$ is $p-1$. On the other hand for each fixed $x$ the
number of $y$ satisfying $y^{2}=x^{2}+t\pmod{p}$ is
$1+\genfrac{(}{)}{}{}{x^{2}+t}{p}$. Summing this number over the $p$
$x$-values and equating the result to $p-1$ produces the required result. ∎

######  Corollary 4.3.

The number of $(x,y)$ solutions to the congruence $x^{2}+y^{2}=0\pmod{p}$ is

$\begin{cases}2p-1&\text{if }\,\,p=1\pmod{4}\\\
1&\text{if}\,\,\,p=3\pmod{4}.\end{cases}$

For each non-zero $t\in\mathbb{Z}_{p}$, the number of $(x,y)$ solutions to the
congruence $x^{2}+y^{2}=t\pmod{p}$ is

$\begin{cases}p-1&\text{if }\,\,p=1\pmod{4}\\\
p+1&\text{if}\,\,\,p=3\pmod{4}.\end{cases}$

###### Proof.

Firstly, $-1$ is a quadratic residue $\pmod{p}\iff p=1\pmod{4}$. Therefore,
$x^{2}=-y^{2}$ has no solution if $p=3\pmod{4}$ other than $(0,0)$ and has
$2(p-1)+1=2p-1$ solutions when $p=1\pmod{4}$. Next, if $t\neq 0$, the number
of $(x,y)$ satisfying $x^{2}+y^{2}=t\pmod{p}$ is

$\sum_{x=0}^{p-1}(1+\genfrac{(}{)}{}{}{t-x^{2}}{p})$
$=p+\sum_{x=0}^{p-1}\genfrac{(}{)}{}{}{t-x^{2}}{p}$
$=p+\genfrac{(}{)}{}{}{-1}{p}\sum_{x=0}^{p-1}\genfrac{(}{)}{}{}{x^{2}-t}{p}.$

The corollary follows from Lemma 4.2. ∎

In terms of the vector $\gamma$, Corollary 4.3 says that when $p=1\pmod{4}$,

$\gamma_{2}=\begin{pmatrix}2p-1\\\ p-1\\\ p-1\end{pmatrix}$ (5)

and when $p=3\pmod{4}$,

$\gamma_{2}=\begin{pmatrix}1\\\ p+1\\\ p+1\end{pmatrix}.$ (6)

This establishes theorems 1.6 and 1.8 for $N_{2}$.

In order to establish a recursive formula for the vector $\gamma_{m}$ in terms
of $\gamma_{m-1}$ we need two preliminary results.

When $t$ is a non-residue mod $p$ we have:

######  Lemma 4.4.

Suppose $\genfrac{(}{)}{}{}{t}{p}=-1$. Then

$\lvert\Big{\\{}z:\genfrac{(}{)}{}{}{t-z^{2}}{p}=-1\Big{\\}}\rvert=\begin{cases}\frac{p+1}{2}&\text{if
}\,\,p=1\pmod{4}\\\ \frac{p-1}{2}&\text{if}\,\,p=3\pmod{4}\end{cases}$

and

$\lvert\Big{\\{}z:\genfrac{(}{)}{}{}{t-z^{2}}{p}=1\Big{\\}}\rvert=\begin{cases}\frac{p-1}{2}&\text{if
}\,\,p=1\pmod{4}\\\ \frac{p+1}{2}&\text{if}\,\,\,p=3\pmod{4}\end{cases}$

###### Proof.

Since $t$ is a non-residue, $t-z^{2}\neq 0\pmod{p}$ for all $z$. From
Corollary 4.3, the equation $x^{2}+z^{2}=t$ has $p-1$ solutions $(x,z)$ if
$p=1\pmod{4}$ and $p+1$ solutions $(x,z)$ if $p=3\pmod{4}$. Therefore,
$t-z^{2}$ is a quadratic residue $\pmod{p}$ for $\frac{p-1}{2}$ values of $z$
if $p=1\pmod{4}$ and for $\frac{p+1}{2}$ values of $z$ if $p=3\pmod{4}$. Since
$t-z^{2}\neq 0\pmod{p}$, $t-z^{2}$ is a non-quadratic residue $\pmod{p}$ for
the remaining values of $z$. ∎

When $t$ is a residue mod $p$ we have:

######  Lemma 4.5.

Suppose $\genfrac{(}{)}{}{}{t}{p}=1$. Then

$\lvert\Big{\\{}z:\genfrac{(}{)}{}{}{t-z^{2}}{p}=-1\Big{\\}}\rvert=\begin{cases}\frac{p-1}{2}&\text{if
}\,\,p=1\pmod{4}\\\ \frac{p-3}{2}&\text{if}\,\,p=3\pmod{4}\end{cases}$

and

$\lvert\Big{\\{}z:\genfrac{(}{)}{}{}{t-z^{2}}{p}=1\Big{\\}}\rvert=\begin{cases}\frac{p-3}{2}&\text{if
}\,\,p=1\pmod{4}\\\ \frac{p-1}{2}&\text{if}\,\,p=3\pmod{4}\end{cases}$

###### Proof.

Since $t$ is a residue, $t-z^{2}=0\pmod{p}$ when $z=\pm\sqrt{t}$. When
$p=1\pmod{4}$, Corollary 4.3 says there are $p-1$ solutions $(x,z)$ to the
congruence $x^{2}+z^{2}=t$ and these solutions can be written as:

$(0,\pm\sqrt{t}),(\pm\sqrt{t},0),(x_{i},z_{i})_{i=1}^{p-5}$

where the solutions are arranged so that $z_{2i-1}=z_{2i}$ for all $i$ (i.e
$x_{2i-1}=-x_{2i}$). Excluding the solutions $(0,\pm\sqrt{t})$ and including
$z=0$, we see that $t-z^{2}$ is a quadratic residue for
$1+\frac{p-5}{2}=\frac{p-3}{2}$ values of $z$ when $p=1\pmod{4}$. A similar
argument can be used when $p=3\pmod{4}$. ∎

Let $t\in\mathbb{Z}_{p}$. By writing the congruence
$\sum_{i=1}^{m+1}{x_{i}}^{2}=t\,\,$ as
$\,\,\sum_{i=1}^{m}{x_{i}}^{2}=t-x_{m+1}^{2}$, we have:

$\displaystyle\begin{split}N_{m+1}(t,p)&=\sum_{s\in\mathbb{Z}_{p}}\lvert\Big{\\{}(x_{i})_{i=1}^{m}:\sum_{i=1}^{m}{x_{i}}^{2}=s\Big{\\}}\rvert\times\lvert\Big{\\{}x_{m+1}:t-x_{m+1}^{2}=s\Big{\\}}\rvert\\\
&=\lvert\Big{\\{}(x_{i})_{i=1}^{m}:\sum_{i=1}^{m}{x_{i}}^{2}=0\Big{\\}}\rvert\times\lvert\Big{\\{}x_{m+1}:t-x_{m+1}^{2}=0\Big{\\}}\rvert\\\
&+\,\,\lvert\Big{\\{}(x_{i})_{i=1}^{m}:\sum_{i=1}^{m}{x_{i}}^{2}=s_{1}\Big{\\}}\rvert\times\lvert\Big{\\{}x_{m+1}:\genfrac{(}{)}{}{}{t-x_{m+1}^{2}}{p}=-1\Big{\\}}\rvert\\\
&+\,\,\lvert\Big{\\{}(x_{i})_{i=1}^{m}:\sum_{i=1}^{m}{x_{i}}^{2}=s_{2}\Big{\\}}\rvert\times\lvert\Big{\\{}x_{m+1}:\genfrac{(}{)}{}{}{t-x_{m+1}^{2}}{p}=1\Big{\\}}\rvert\end{split}$
(7)

where $s_{1}$ is a fixed non-residue $\pmod{p}$ and $s_{2}$ is a fixed residue
$\pmod{p}$ (the particular choices do not matter due to Lemma 2.1). We can
rewrite equation (7) as

$\displaystyle\begin{split}N_{m+1}(t,p)={}&\gamma_{m,1}\times\lvert\\{x_{m+1}:t-x_{m+1}^{2}=0\\}\rvert\,\,+\\\
&\gamma_{m,2}\times\lvert\\{x_{m+1}:\genfrac{(}{)}{}{}{t-x_{m+1}^{2}}{p}=-1\\}\rvert\,\,+\\\
&\gamma_{m,3}\times\lvert\\{x_{m+1}:\genfrac{(}{)}{}{}{t-x_{m+1}^{2}}{p}=1\\}\rvert.\end{split}$
(8)

By substituting in turn $t=0$, $t$ a non-residue and $t$ a residue $\pmod{p}$
in equation (8), we obtain three linear equations for the components of
$\gamma_{m+1}$ in terms of the components of $\gamma_{m}$. The resulting
matrix equation can then be solved. Obviously,

$\lvert\\{x_{m+1}:-x_{m+1}^{2}=0\\}\rvert=1$

and $\lvert\\{x_{m+1}:\genfrac{(}{)}{}{}{-x_{m+1}^{2}}{p}=\pm 1\\}\rvert$ is
either $0$ or $p-1$ depending on $p\pmod{4}$. We can obtain
$\lvert\\{x_{m+1}:\genfrac{(}{)}{}{}{t-x_{m+1}^{2}}{p}=-1\\}\rvert$ and
$\lvert\\{x_{m+1}:\genfrac{(}{)}{}{}{t-x_{m+1}^{2}}{p}=1\\}\rvert$ from Lemma
4.4 and Lemma 4.5. The resulting equations can be represented in matrix form
as $\gamma_{m+1}=A\times\gamma_{m}$ when $p=1\pmod{4}$, and
$\gamma_{m+1}=B\times\gamma_{m}$ when $p=3\pmod{4}$, where the matrices $A$
and $B$ are defined by:

$\displaystyle A:=\begin{pmatrix}1,0,p-1\\\ 0,\frac{p+1}{2},\frac{p-1}{2}\\\
2,\frac{p-1}{2},\frac{p-3}{2}\end{pmatrix}\,\,\text{ and
}\,\,B:=\begin{pmatrix}1,p-1,0\\\ 0,\frac{p-1}{2},\frac{p+1}{2}\\\
2,\frac{p-3}{2},\frac{p-1}{2}\end{pmatrix}.$ (9)

Therefore,

$\gamma_{m}=A^{m-1}\times\gamma_{1}\,\,\text{ if p = 1 mod 4}$

and

$\gamma_{m}=B^{m-1}\times\gamma_{1}\,\,\text{ if p = 3 mod 4}.$

Theorems 1.5 and 1.6 now follow from the form of $\gamma_{1}$ and $\gamma_{2}$
and the identities:

$A^{2}\times\begin{pmatrix}p^{2m}\\\ p^{2m}-p^{m}\\\
p^{2m}+p^{m}\end{pmatrix}=\begin{pmatrix}p^{2m+2}\\\ p^{2m+2}-p^{m+1}\\\
p^{2m+2}+p^{m+1}\end{pmatrix}\,\,\text{for}\,\,m\geq 0,$
$A^{2}\times\begin{pmatrix}p^{2m-1}+p^{m}-p^{m-1}\\\ p^{2m-1}-p^{m-1}\\\
p^{2m-1}-p^{m-1}\end{pmatrix}=\begin{pmatrix}p^{2m+1}+p^{m+1}-p^{m}\\\
p^{2m+1}-p^{m}\\\ p^{2m+1}-p^{m}\end{pmatrix}\,\,\text{for}\,\,m\geq 1.$

Similarly, Theorems 1.7 and 1.8 follow from the identities:

$B^{2}\times\begin{pmatrix}p^{2m}\\\ p^{2m}+(-1)^{m+1}p^{m}\\\
p^{2m}+(-1)^{m}p^{m}\end{pmatrix}=\begin{pmatrix}p^{2m+2}\\\
p^{2m+2}+(-1)^{m}p^{m+1}\\\
p^{2m+2}+(-1)^{m+1}p^{m+1}\end{pmatrix}\,\,\text{for}\,\,m\geq 0,$

and

$\displaystyle\begin{split}B^{2}\times&\begin{pmatrix}p^{2m-1}+(-1)^{m}p^{m}+(-1)^{m-1}p^{m-1}\\\
p^{2m-1}+(-1)^{m-1}p^{m-1}\\\ p^{2m-1}+(-1)^{m-1}p^{m-1}\end{pmatrix}\\\ \\\
=&\begin{pmatrix}p^{2m+1}+(-1)^{m+1}p^{m+1}+(-1)^{m}p^{m}\\\
p^{2m+1}+(-1)^{m}p^{m}\\\
p^{2m+1}+(-1)^{m}p^{m}\end{pmatrix}\,\,\text{for}\,\,m\geq 1.\end{split}$

##  5 Sum of two squares in $\mathbb{Z}_{p^{k}}$: $N_{2}(t,p^{k})$

###  5.1 Proof of theorem 1.9: $p=1\pmod{4}$

We first deal with primes $p:p=1\pmod{4}$ and calculate $N_{2}$ in
$\mathbb{Z}_{p^{k}}$ where $k\geq 1$.

Since $\genfrac{(}{)}{}{}{-1}{p}=1$, $\sqrt{-1}$ exists in
$\mathbb{Z}_{p^{k}}$ by Hensel’s Lemma. We make the invertible transformation:

$u=x+y\,\sqrt{-1}\,,\,\,\,\,v=x-y\,\sqrt{-1}.$

Then, for $t\in\mathbb{Z}_{p^{k}}$:

$\lvert\\{(x,y):x^{2}+y^{2}=t\pmod{p^{k}}\\}\rvert\,\,=\,\,\lvert\\{(u,v):uv=t\pmod{p^{k}}\\}\rvert.$

Let $t=0$. Then, $uv=0$ if and only if $u=0$ or $v=0$ or there is an
$0\leq\alpha<k$ such that $p^{\alpha}\|u$ and $p^{k-\alpha}|v$. We have

$\lvert\\{u:p^{\alpha}\|u\\}\rvert=p^{k-\alpha-1}(p-1)\,\,\text{ and
}\,\,\lvert\\{v:p^{k-\alpha}|v\\}\rvert=p^{\alpha}-1.$

So,

$\displaystyle\lvert\\{(u,v):uv=0\pmod{p^{k}}\\}\rvert$
$\displaystyle=2p^{k}-1+\sum_{\alpha=0}^{k-1}p^{k-\alpha-1}(p^{\alpha}-1)(p-1)$
$\displaystyle=p^{k-1}\left(p(k+1)-k\right).$

Next, assume $p^{\alpha}\|t$ and write $t=p^{\alpha}\beta$ where
$0\leq\alpha<k$ and $p\nmid\beta$. Then, $uv=t$ if and only if
$u=p^{\theta}\delta$ and $v=p^{\alpha-\theta}\beta\delta^{-1}+np^{k-\theta}$
for some $0\leq\theta\leq\alpha$, $0\leq n<p^{k-\theta}$ and
$1\leq\delta<p^{k-\theta}$ with $p\nmid\delta$. Then,

$\displaystyle\lvert\\{(u,v):uv=t\\}\rvert$
$\displaystyle=\sum_{\theta=0}^{\alpha}\lvert\Big{\\{}\delta\in\mathbb{Z}_{p^{k}}:1\leq\delta<p^{k-\theta}\,\,\text{
and }\,\,p\nmid\delta\Big{\\}}\times p^{\theta}\rvert$
$\displaystyle=\sum_{\theta=0}^{\alpha}p^{k-\theta-1}(p-1)p^{\theta}$
$\displaystyle=\left(\alpha+1\right)(p-1)p^{k-1}.$

This completes the proof of theorem 1.9.

###  5.2 Proof of theorem 1.10: $p=3\pmod{4}$

In this section we assume $p=3\pmod{4}$ and determine $N_{2}$ in
$\mathbb{Z}_{p^{k}}$ where $k\geq 1$.

We will first look at $N_{2}(0,p^{k})$. When $p=3\pmod{4}$, there are no non-
trivial solutions $(x,y)$ to $x^{2}+y^{2}=0\pmod{p}$. Therefore,

$ord_{p}(x^{2}+y^{2})=\min\\{ord_{p}(x^{2}),ord_{p}(y^{2})\\}.$ (10)

So if $x^{2}+y^{2}=0\pmod{p^{k}}$, both $x^{2}$ and $y^{2}$ must be zero
$\pmod{p^{k}}$. We have,

$\\{x:x^{2}=0\pmod{p^{k}}\\}=\\{0,p^{\lceil\frac{k}{2}\rceil},2p^{\lceil\frac{k}{2}\rceil},\dots,(p^{k-\lceil\frac{k}{2}\rceil}-1)p^{\lceil\frac{k}{2}\rceil}\\}.$

Hence,

$\lvert\\{x:x^{2}=0\pmod{p^{k}}\\}\rvert=p^{k-\lceil\frac{k}{2}\rceil}=p^{\lfloor\frac{k}{2}\rfloor}$

and the same result holds for $y$. Therefore,
$N_{2}(0,p^{k})=p^{2\lfloor\frac{k}{2}\rfloor}$.

Next let $t\in\mathbb{Z}_{p^{k}}$ and assume $p^{2\alpha+1}\|t$ for some
$\alpha:1\leq 2\alpha+1<k$. Then $x^{2}+y^{2}\neq t\pmod{p^{k}}$ from (10).
Hence, $N_{2}(t,p^{k})=0$ in this case.

Next assume $p\nmid t$. Any pair $(x,y)$ satisfying
$x^{2}+y^{2}=t\pmod{p^{k}}$ is of the form $x=x_{1}+mp^{k-1}$,
$y=y_{1}+np^{k-1}$ where ${x_{1}}^{2}+{y_{1}}^{2}=t\pmod{p^{k-1}}$ and $0\leq
m,n<p$. Then ${x_{1}}^{2}+{y_{1}}^{2}=t+rp^{k-1}$ for some $r$. Expanding the
initial congruence we find that $p$ must divide $2x_{1}m+2y_{1}n+r$. Since
$p\nmid t$, one of $x_{1}$ or $y_{1}$ (say $x_{1}$) is not divisible by $p$
and is thus invertible in $\mathbb{Z}_{p}$. If $y_{1}=0$, then
$m=-\frac{r}{2x_{1}}\pmod{p}$ and $n=\\{0,1,\dots,p-1\\}$. If $y_{1}\neq 0$,
$m=\frac{-r-2y_{1}n}{2x_{1}}\pmod{p}$. In either case, there are $p$ choices
for the pair $(m,n)$ producing a solution to the original congruence.
Therefore,

$\lvert\\{(x,y):x^{2}+y^{2}=t\pmod{p^{k}}\\}\rvert=p\times\lvert\\{(x,y):x^{2}+y^{2}=t\pmod{p^{k-1}}\\}\rvert.$

From theorem 1.8, $\lvert\\{(x,y):x^{2}+y^{2}=t\pmod{p}\\}\rvert=p+1$. So,
when $p\nmid t$,

$\lvert\\{(x,y):x^{2}+y^{2}=t\pmod{p^{k}}\\}\rvert=(p+1)p^{k-1}$ (11)

We now assume $p^{2\alpha}\|t$ for some $\alpha:0<2\alpha<k$ and write
$t=p^{2\alpha}\beta$ where $p\nmid\beta$. If, $x^{2}+y^{2}=t\pmod{p^{k}}$,
then $x^{2}+y^{2}=0\pmod{p}$ and so $x,y=0\pmod{p}$. Putting $x=px_{1}$,
$y=py_{1}$ and dividing the congruence through by $p^{2}$ we have
${x_{1}}^{2}+{y_{1}}^{2}=p^{2\alpha-2}\beta\pmod{p^{k-2}}$. Continuing in this
way, we find $m,n\in\mathbb{Z}_{p^{k-\alpha}}$ such that $x=p^{\alpha}m$,
$y=p^{\alpha}n$ and $m^{2}+n^{2}=\beta\pmod{p^{k-2\alpha}}$. From (11) there
are $(p+1)p^{k-2\alpha-1}$ solutions to the congruence
$a^{2}+b^{2}=\beta\pmod{p^{k-2\alpha}}$. Each of these solutions $(a,b)$
generates $p^{\alpha}$ values for $m,n\in\mathbb{Z}_{p^{k-\alpha}}$ given by

$m\in\\{a,a+p^{k-2\alpha},\dots,a+(p^{\alpha}-1)p^{k-2\alpha}\\},$
$n\in\\{b,b+p^{k-2\alpha},\dots,b+(p^{\alpha}-1)p^{k-2\alpha}\\}.$

The number of solutions to $x^{2}+y^{2}=p^{\alpha}\beta\pmod{p^{k}}$ is

$(p+1)p^{k-2\alpha-1}\times p^{\alpha}\times p^{\alpha}=(p+1)p^{k}.$

This completes the proof of theorem 1.10.

##  6 Sum of three squares in $\mathbb{Z}_{p^{k}}$: $N_{3}(t,p^{k})$

###  6.1 Some preliminary lemmas

In this section we will calculate $N_{3}$ in $\mathbb{Z}_{p^{k}}$ where $k\geq
1$. We will use the results for $r_{2}$ and the decomposition:

$N_{3}(t,p^{k})=\sum_{s\in\mathbb{Z}_{p^{k}}}N_{2}(s,p^{k})\times\lvert\Big{\\{}z:t-z^{2}=s\Big{\\}}\rvert.$
(12)

Let $t=p^{\alpha}\beta\in\mathbb{Z}_{p^{k}}$ where $p\nmid\beta$. The
following lemmas give the number of $z\in\mathbb{Z}_{p^{k}}\,\,$ satisfying
$\,\,p^{\gamma}\|t-z^{2}\,\,$ for each $\gamma:0\leq\gamma<k$.

######  Lemma 6.1.

$\lvert\Big{\\{}z:z^{2}=0\pmod{p^{k}}\Big{\\}}\rvert=p^{\lfloor k/2\rfloor}.$

###### Proof.

$z^{2}=0\,\,$ if and only if $z=0$ or $z=sp^{\lceil k/2\rceil}$ for some
$\,\,s:1\leq s<p^{k-\lceil k/2\rceil}-1$. ∎

######  Lemma 6.2.

If $t\neq 0$,

$\lvert\Big{\\{}z:t-z^{2}=0\pmod{p^{k}}\Big{\\}}\rvert=\begin{cases}2p^{\alpha/2}&\text{if
}\,\,\alpha\text{ is even and }\genfrac{(}{)}{}{}{\beta}{p}=1\\\ 0&\text{
otherwise}.\end{cases}$

###### Proof.

If $\genfrac{(}{)}{}{}{t}{p}=1$, $\sqrt{t}$ exists in $\mathbb{Z}_{p^{k}}$.
Then $t-z^{2}=0\pmod{p^{k}}\,\,$ if and only if
$z=\pm\sqrt{t}+sp^{k-\alpha/2}$ for some $\,\,s:0\leq s<p^{\alpha/2}-1$. ∎

######  Lemma 6.3.

If $\alpha$ is odd and $\gamma:0\leq\gamma<k$ then

$\lvert\Big{\\{}z:p^{\gamma}\|t-z^{2}\pmod{p^{k}}\Big{\\}}\rvert=\begin{cases}(p-1)p^{k-1-\gamma/2}&\text{if
}\,\,\gamma\text{ is even and }0\leq\gamma<\alpha\\\
p^{k-(\alpha+1)/2}&\text{if }\,\,\gamma=\alpha\\\ 0&\text{
otherwise}.\end{cases}$

###### Proof.

If $z=0\,\,$, $p^{\alpha}\|t-z^{2}$. Suppose $z\neq 0$ with $p^{\delta}\|z$.
Then,

$\displaystyle\begin{split}p^{2\delta}\|t-z^{2}&\,\,\text{ for
}0\leq\delta<\frac{\alpha}{2}\\\ p^{\alpha}\|t-z^{2}&\,\,\text{ for
}\frac{\alpha}{2}<\delta<k.\end{split}$ (13)

The statement for $\gamma$ even and less than $\alpha$ then follows from

$\lvert\\{z:p^{\delta}\|z\\}\rvert=(p-1)p^{k-1-\delta}.$

The statement for $\gamma=\alpha$ follows from

$\displaystyle\lvert\\{z:p^{\delta}\|z\,\,\text{ for
}\frac{\alpha}{2}<\delta<k\\}\rvert$
$\displaystyle=\sum_{\delta=(\alpha+1)/2}^{k-1}(p-1)p^{k-1-\delta}$
$\displaystyle=\,\,p^{k-(\alpha+1)/2}-1.$

If $\gamma$ is odd but not equal to $\alpha$, it is clear from (13) that
$ord_{p}(t-z^{2})\neq\gamma$. ∎

######  Lemma 6.4.

If $\alpha$ is even, $\genfrac{(}{)}{}{}{\beta}{p}=-1$ and
$\gamma:0\leq\gamma<k$ then

$\lvert\Big{\\{}z:p^{\gamma}\|t-z^{2}\pmod{p^{k}}\Big{\\}}\rvert=\begin{cases}(p-1)p^{k-1-\gamma/2}&\text{if
}\,\,\gamma\text{ is even and }0\leq\gamma<\alpha\\\ 0&\text{if
}\,\,\gamma\text{ is odd }\\\ p^{k-\alpha/2}&\text{if }\,\,\gamma=\alpha\\\
0&\text{ if }\,\,\alpha<\gamma<k.\end{cases}$

###### Proof.

If $z=0\,\,$, $p^{\alpha}\|t-z^{2}$. Suppose $z\neq 0$ with
$z=p^{\delta}\epsilon$. Then, (13) holds and the statement for $\gamma$ even
and less than $\alpha$ follows in the same way as for lemma 6.3. When
$\delta=\alpha/2$, $t-z^{2}=p^{\alpha}(\beta-\epsilon^{2})$. As
$\genfrac{(}{)}{}{}{\beta}{p}=-1$, $\beta-\epsilon^{2}\neq 0\pmod{p}$ and so
$p^{\alpha}\|t-z^{2}$. Therefore, $p^{\alpha}\|t-z^{2}$ when
$\alpha/2\leq\delta<k$. The statement for the case $\gamma=\alpha$ follows
from:

$\displaystyle\lvert\\{z:p^{\delta}\|z\,\,\text{ for
}\frac{\alpha}{2}\leq\delta<k\\}\rvert$
$\displaystyle=\sum_{\delta=\alpha/2}^{k-1}(p-1)p^{k-1-\delta}=\,\,p^{k-\alpha/2}-1.$

If $\gamma$ is odd or $\gamma>\alpha$, it is clear from (13) that
$ord_{p}(t-z^{2})\neq\gamma$. ∎

######  Lemma 6.5.

If $\alpha$ is even, $\genfrac{(}{)}{}{}{\beta}{p}=1$ and
$\gamma:0\leq\gamma<k$ then

$\lvert\Big{\\{}z:p^{\gamma}\|t-z^{2}\pmod{p^{k}}\Big{\\}}\rvert=\begin{cases}(p-1)p^{k-1-\gamma/2}&\text{if
}\,\,\gamma\text{ is even and }0\leq\gamma<\alpha\\\ 0&\text{if
}\,\,\gamma\text{ is odd and }0\leq\gamma<\alpha\\\
(p-2)p^{k-1-\alpha/2}&\text{if }\,\,\gamma=\alpha\\\
2(p-1)p^{k-1+\alpha/2-\gamma}&\text{ if }\,\,\alpha<\gamma<k.\end{cases}$

###### Proof.

Suppose $z\neq 0$ and write $z=p^{\delta}\epsilon$ with $p\nmid\epsilon$ and
$1\leq\epsilon<p^{k-\delta}$.

When $\gamma$ is even and $0\leq\gamma<\alpha$, the calculation is the same as
for lemmas 6.3 and 6.4 above.

If $\gamma$ is odd and less than $\alpha$, (13) shows there is no $z$ with
$p^{\gamma}\|t-z^{2}$.

Now, $p^{\alpha}\|t-z^{2}$ if and only if $z=0$ or $p^{\alpha/2+1}|z$ or
$p^{\alpha/2}\|z$ and $p\nmid\beta-\epsilon^{2}$. We have:

$\lvert\\{z:p^{\alpha/2+1}|z\\}\rvert=p^{k-1-\alpha/2}-1$

and

$\displaystyle\lvert\\{z:p^{\alpha/2}\|z\,\,\text{and}\,\,p\nmid\beta-\epsilon^{2}\\}\rvert$
$\displaystyle=\lvert\\{\epsilon:1\leq\epsilon<p^{k-\alpha/2},\,\,p\nmid\epsilon\,\,\text{and}\,\,p\nmid\beta-\epsilon^{2}\\}\rvert$
$\displaystyle=(p-1)p^{k-1-\alpha/2}-2p^{k-1-\alpha/2}$

since $\,\,p|\beta-\epsilon^{2}$ if and only if $\epsilon=\pm\sqrt{\beta}+np$
for some $n$ with $0\leq n<p^{k-1-\alpha/2}$. Adding the three components
gives the result for $\gamma=\alpha$.

If $\gamma>\alpha$, then $p^{\gamma}\|t-z^{2}$ if and only if
$p^{\alpha/2}\|z$ and $p^{\gamma-\alpha}\|\beta-\epsilon^{2}$. We have:

$\displaystyle\lvert\\{z:p^{\alpha/2}\|z\,\,\text{and}\,\,p^{\gamma-\alpha}\|\beta-\epsilon^{2}\\}\rvert$
$\displaystyle=\lvert\\{\epsilon:1\leq\epsilon<p^{k-\alpha/2},\,\,p\nmid\epsilon,\,\,p^{\gamma-\alpha}\|\beta-\epsilon^{2}\\}\rvert$
$\displaystyle=2(p-1)p^{k-1+\alpha/2-\gamma}$

since $p^{\gamma-\alpha}\|\beta-\epsilon^{2}$ if and only if
$\epsilon=\pm\sqrt{\beta}+np^{\gamma-\alpha}$ where $p\nmid n$ and $0\leq
n<p^{k+\alpha/2-\gamma}$.

∎

###  6.2 Proof of theorem 1.11 and theorem 1.12

Let $t\in\mathbb{Z}_{p^{k}}$ with $t=p^{\alpha}\beta$. Proving theorems 1.11
and 1.12 requires a separate calculation for each of the possibilities for
$t$. The four possibilities for $t$ are: $t=0$, $\alpha$ odd, $\alpha$ even
and $\genfrac{(}{)}{}{}{\beta}{p}=1$ and $\alpha$ even and
$\genfrac{(}{)}{}{}{\beta}{p}=-1$. In addition we need to consider the value
of $p\pmod{4}$. We provide the calculations for a few of the eight possible
cases. The calculations for the other cases follow the same pattern.

####  6.2.1 The case $p=1\pmod{4}$ and $t=0$

When $t=0$, equation (12) becomes:

$N_{3}(0,p^{k})=\sum_{s\in\mathbb{Z}_{p^{k}}}N_{2}(s,p^{k})\times\lvert\Big{\\{}z:-z^{2}=s\Big{\\}}\rvert.$
(14)

Writing each $s$ (other than $s=0$) in the sum in equation (14) as
$s=p^{\gamma}b$ where $p\nmid b$, we can use theorem 1.9 and lemma 6.2, to
get:

$N_{3}(0,p^{k})=p^{k-1}\left(p(k+1)-k\right)*p^{\lfloor
k/2\rfloor}+\sum_{\gamma\,\,\text{even}}\,\,\sum_{b:\genfrac{(}{)}{}{}{b}{p}=-1}(\gamma+1)(p-1)p^{k-1}*2p^{\gamma/2}.$

For fixed $\gamma$, $\lvert\\{s:p^{\gamma}\|s\text{ and
}\genfrac{(}{)}{}{}{b}{p}=-1\\}\rvert=\frac{1}{2}(p-1)p^{k-\gamma-1}$.
Therefore,

$\displaystyle N_{3}(0,p^{k})$
$\displaystyle=p^{k-1}\left(p(k+1)-k\right)*p^{\lfloor
k/2\rfloor}+(p-1)^{2}p^{k-1}\sum_{\gamma\,\,\text{even}}(\gamma+1)p^{k-\gamma/2-1}$
$\displaystyle=p^{k-1}\left(p(k+1)-k\right)*p^{\lfloor
k/2\rfloor}+(p-1)^{2}p^{k-1}\sum_{\gamma=0}^{\lfloor(k-1)/2\rfloor}(2\gamma+1)p^{k-\gamma-1}$

after reindexing. The two parts of the sum are evaluated as:

$(p-1)^{2}p^{k-1}\sum_{\gamma=0}^{\lfloor(k-1)/2\rfloor}\gamma
p^{k-\gamma-1}=p^{2k-1}-\lfloor\frac{k+1}{2}\rfloor
p^{2k-\lfloor(k+1)/2\rfloor}+\lfloor\frac{k-1}{2}\rfloor
p^{2k-1-\lfloor(k+1)/2\rfloor}$

and

$(p-1)^{2}p^{k-1}\sum_{\gamma=0}^{\lfloor(k-1)/2\rfloor}p^{k-\gamma-1}=p^{2k}-p^{2k-1}-p^{2k-\lfloor(k+1)/2\rfloor}+p^{2k-1-\lfloor(k+1)/2\rfloor}.$

Manipulating the $floor$ and $ceiling$ functions produces the result for $t=0$
in theorem 1.11.

####  6.2.2 The case $p=3\pmod{4}$ and $t=0$

Now consider the case $p=3\pmod{4}$. Using theorem 1.10 and lemma 6.2:

$\displaystyle N_{3}(0,p^{k})$ $\displaystyle=p^{2\lfloor
k/2\rfloor}*p^{\lfloor k/2\rfloor}+\sum_{s\neq 0}N_{2}(s,p^{k})*2p^{\gamma}$
$\displaystyle=p^{3\lfloor k/2\rfloor}+\sum_{s\neq 0}(p+1)p^{k-1}*2p^{\gamma}$

where the sums in the two lines above are over $s:s=p^{2\gamma}b$ with $0\leq
2\gamma<k$, $p\nmid b$, $\genfrac{(}{)}{}{}{b}{p}=-1$ and $1\leq
b<p^{k-2\gamma}$. For fixed $\gamma$, the number of $b$ satisfying the last
three conditions is $\frac{1}{2}p^{k-2\gamma-1}(p-1)$. We therefore have:

$\displaystyle N_{3}(0,p^{k})=$ $\displaystyle p^{3\lfloor
k/2\rfloor}+\sum_{\gamma=0}^{\lfloor(k-1)/2\rfloor}\frac{1}{2}p^{k-2\gamma-1}(p-1)(p+1)p^{k-1}*2p^{\gamma}$
$\displaystyle=$ $\displaystyle p^{3\lfloor
k/2\rfloor}+(p+1)p^{k-1}\sum_{\gamma=0}^{\lfloor(k-1)/2\rfloor}(p-1)p^{k-\gamma-1}$
$\displaystyle=$ $\displaystyle p^{3\lfloor
k/2\rfloor}+(p+1)p^{k-1}\left(p^{k}-p^{k-1-\lfloor(k-1)/2\rfloor}\right)$
$\displaystyle=$ $\displaystyle p^{2k}+p^{2k-1}+p^{3\lfloor
k/2\rfloor}-(p+1)p^{2k-2-\lfloor(k-1)/2\rfloor}.$

This is equivalent to the statement for $t=0$ in theorem 1.12.

####  6.2.3 The case $p=1\pmod{4}$ and $\alpha$ odd

Theorem 1.9 shows that for $s\in\mathbb{Z}_{p^{k}}$, $N_{2}(s,p^{k})$ depends
only on $ord_{p}(s)$. We can use (12) and lemmas 6.2 and 6.3 in this case to
get:

$\displaystyle\begin{split}N_{3}(t,p^{k})=\,\,&p^{k-1}\left(p(k+1)-k\right)\times\lvert\Big{\\{}z:t-z^{2}=0\Big{\\}}\rvert\\\
&\,\,+\,\,\sum_{\gamma=0}^{k-1}(\gamma+1)(p-1)p^{k-1}\times\lvert\Big{\\{}z:p^{\gamma}\|t-z^{2}\Big{\\}}\rvert\\\
=&\sum_{\gamma=0}^{(\alpha-1)/2}(2\gamma+1)(p-1)p^{k-1}\times(p-1)p^{k-1-\gamma}+(\alpha+1)(p-1)p^{k-1}\times
p^{k-(\alpha+1)/2}\end{split}$

where the sum has been reindexed to account for even $\gamma$. The two parts
of the sum are evaluated as:

$2(p-1)^{2}p^{2k-2}\sum_{\gamma=0}^{(\alpha-1)/2}\gamma
p^{-\gamma}=2p^{2k-1}-\left(\alpha+1\right)p^{2k-(\alpha+1)/2}+\left(\alpha-1\right)p^{2k-(\alpha+3)/2}$

and

$(p-1)^{2}p^{2k-2}\sum_{\gamma=0}^{(\alpha-1)/2}p^{-\gamma}=(p-1)(p^{2k-1}-p^{2k-(\alpha+3)/2}).$

The formula in theorem 1.11 for the case when $\alpha$ is odd follows.

####  6.2.4 The case $p=1\pmod{4},\,\,\alpha$ even and
$\genfrac{(}{)}{}{}{\beta}{p}=1$

We use (12), theorem 1.9, lemmas 6.2 and 6.5 in this case to get:

$\displaystyle N_{3}(t,p^{k})$
$\displaystyle=\Sigma_{1}+\Sigma_{2}+\Sigma_{3}+\Sigma_{4}$

where, $\Sigma_{1}$ is the contribution to $N_{3}(t,p^{k})$ in (12) from
$s=0$, $\Sigma_{2}$ is the contribution from $s$ such that $0\leq
ord_{p}(s)<\alpha$, $\Sigma_{3}$ comes from $ord_{p}(s)=\alpha$ and
$\Sigma_{4}$ comes from $s$ with $\alpha<ord_{p}(s)<k$. So,

$\displaystyle\Sigma_{1}=p^{k-1}\Big{(}p(k+1)-k\Big{)}\times 2p^{\alpha/2}.$

After re-indexing for even $\gamma$ in $\Sigma_{2}$,

$\displaystyle\Sigma_{2}$
$\displaystyle=(p-1)p^{k-1}\sum_{\gamma=0}^{\alpha/2-1}(2\gamma+1)(p-1)p^{k-1-\gamma}$
$\displaystyle=p^{2k}+p^{2k-1}-(\alpha+1)p^{2k-\alpha/2}+(\alpha-1)p^{2k-1-\alpha/2}.$

We also have:

$\displaystyle\Sigma_{3}$
$\displaystyle=(p-1)(p-2)(\alpha+1)p^{2k-2-\alpha/2}$

and

$\displaystyle\Sigma_{4}$
$\displaystyle=2(p-1)^{2}p^{k-1}\sum_{\gamma=\alpha+1}^{k-1}(\gamma+1)p^{k-1+\alpha/2-\gamma}$
$\displaystyle=2(\alpha+2)p^{2k-1-\alpha/2}-2(k+1)p^{k+\alpha/2}+2kp^{k-1+\alpha/2}-2(\alpha+1)p^{2k-2-\alpha/2}.$

Adding up the various terms gives the formula for $N_{3}(t,p^{k})$ in theorem
1.11 when $\alpha$ is even and $\genfrac{(}{)}{}{}{\beta}{p}=1$.

## References

  * [1] Fabián Arias, Jerson Borja, and Luis Rubio.  Counting integers representable as images of polynomials modulo $n$.  arXiv, arXiv:1812.11599, 12 2018.
  * [2] Emil Artin.  Über die Zerlegung definiter Funktionen in Quadrate.  Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 5(1):100–115, Dec 1927.
  * [3] Rob Burns.  Representing numbers as the sum of squares and powers in the ring $\mathbb{Z}_{n}$.  arXiv, arXiv:1708.03930, 08 2017.
  * [4] L. Carlitz.  Solution to problem 140 (proposed by I. Connell).  Canad. Math. Bull., 11:615–619, 1968.
  * [5] Catalina Calderón, José María Grau, Antonio M. Oller-Marcén, and László Tóth.  Counting invertible sums of squares modulo $n$ and a new generalization of Euler’s totient function.  Publicationes Mathematicae Debrecen, 87:133–145, 2015.
  * [6] M.D. Choi, T.Y. Lam, B. Reznick, and A. Rosenberg.  Sums of squares in some integral domains.  Journal of Algebra, 65(1):234 – 256, 1980.
  * [7] James W. Cogdell.  On sums of three squares.  Journal de Théorie des Nombres de Bordeaux, 15(1):33–44, 2003\.
  * [8] Harvey Cohn.  Decomposition into Four Integral Squares in the Fields of $\sqrt{2}$ and $\sqrt{3}$.  American Journal of Mathematics, 82(2), 1960.
  * [9] Jean-Louis Colliot-Thélène and Fei Xu.  Brauer-Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms.  Compositio Mathematica, 145:309–363, 2009.
  * [10] P. Ribenboim D. Gondard.  Le 17e problme de Hilbert pour les matrices.  Bull. Sci. Math., 98:49–56, 1974.
  * [11] L. E. Dickson.  History of the theory of numbers, volume II: Diophantine analysis.  Chelsea Publishing Co, New York, 1966.
  * [12] Leonard Euler.  De numerus qui sunt aggregata quorum quadratorum.  Novi commentarii academiae scientiarum Petropolitanae, 4:3–40, 1752-1753.
  * [13] Leonard Euler.  Demonstratio theorematis Fermatiani omnem numerum primum formae $4n+1$ esse summam duorum quadratorum.  Novi commentarii academiae scientiarum Petropolitanae, 5:3–13, 1754-1755.
  * [14] José F. Fernando, Jesús M. Ruiz, and Claus Scheiderer.  Sums of squares in real rings.  Transactions of the American Mathematical Society, 356(7):2663–2684, 2004.
  * [15] G. H. Hardy and E. M. Wright.  An introduction to the theory of numbers.  Oxford University Press, 5th edition, 1979.
  * [16] Fritz Götzky.  Über eine zahlentheoretische Anwendung von Modulfunktionen zweier Veränderlichen.  Mathematische Annalen, 100:411–437, 1928.
  * [17] José María Grau and Antonio M. Oller-Marcén.  Fast computation of the number of solutions to $x_{1}^{2}+\dots+x_{k}^{2}\equiv\lambda(\mod n)$.  Journal of Number Theory, 200:427–440, 2019.
  * [18] Malcolm Griffin and Mark Krusemeyar.  Matrices as sums of squares.  Linear and Multilinear Algebra, 5(1):33–44, 1977.
  * [19] J Harrington, L Jones, and A Lamarche.  Representing integers as the sum of two squares in the ring $\mathbb{Z}_{n}$.  Journal of Integer Sequences, 17(14.7.4), 2014.
  * [20] Kenneth Ireland and Michael Rosen.  A Classical Introduction to Modern Number Theory.  Springer New York, 1990.
  * [21] Carl Gustav Jacob Jacobi.  Fundamenta nova theoriae functionum ellipticarum.  Borntraeger, Königsberg, 1829.
  * [22] S. A. Katre and Anuradha S. Garge.  Matrices over commutative rings as sums of $k$-th powers.  Proceedings of the American Mathematical Society, 141(1):103–113, May 2012.
  * [23] Edmund Landau.  Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate.  Arch. Math. Phys., 13:304–312, 1908.
  * [24] David B. Leep and Colin L. Starr.  Polynomials in ${R}[x,y]$ that are sums of squares in ${R}(x,y)$.  Proceedings of the American Mathematical Society, 129(11):3133–3141, 2001.
  * [25] Songsong Li and Yi Ouyang.  Counting the solutions of $\lambda_{1}x_{1}^{k_{1}}+\dots+\lambda_{t}x_{t}^{k_{t}}\equiv c(\mod n)$.  Journal of Number Theory, 187:41–65, 2018.
  * [26] Stephen C. Milne.  New Infinite Families of Exact Sums of Squares Formulas, Jacobi Elliptic Functions, and Ramanujan’s Tau Function.  Proceedings of the National Academy of Sciences of the United States of America, 93(26):15004–15008, 1996.
  * [27] Mohsen Mollahajiaghaei.  On the addition of squares of units modulo $n$.  Journal of Number Theory, 170:35–45, jan 2017.
  * [28] Morris Newman.  Sums of squares of matrices.  Pacific J. Math., 118(2):497–506, 1985.
  * [29] Ivan Niven.  Integers of quadratic fields as sums of squares.  Trans. Amer. Math. Soc., 48:405–417, 1940.
  * [30] Ken Ono.  Representations of integers as sums of squares.  Journal of Number Theory, 95:253–258, 2002.
  * [31] Claudio Procesi and Murray Schacher.  A non-commutative real Nullstellensatz and Hilbert’s 17th problem.  Annals of Mathematics, 104(3):395–406, 1976.
  * [32] C. Ramus.  Solution generale d’un probleme d’analyse combinatoire.  J. Reine Ang. Math., 11:352–355, 1834.
  * [33] Rainer Schulze-Pillot.  Representation by integral quadratic forms - a survey.  2004\.
  * [34] Carl Siegel.  Darstellung total positiver Zahlen durch Quadrate.  Mathematische Zeitschrift, 11(3-4):246–275, Sep 1921.
  * [35] Jeff Suzuki.  Euler and Number Theory: A Study in Mathematical Invention.  In Robert E. Bradley and C. Edward Sandifer, editors, Leonhard Euler: Life, Work and Legacy, volume 5 of Studies in the History and Philosophy of Mathematics, pages 363 – 383. Elsevier, 2007.
  * [36] László Tóth.  Counting solutions of quadratic congruences in several variables revisited.  Journal of Integer Sequences, 17(Article 14.11.6), 2014.
  * [37] Quan-Hui Yang and Min Tang.  On the addition of squares of units and nonunits modulo $n$.  Journal of Number Theory, 155:1–12, oct 2015.