A Course in Real Algebraic Geometry:
Positivity and Sums of Squares

This textbook originates from a course for graduate students that I taught at Konstanz University, approximately five or six times over the past twenty years. While the first part of the course remained largely the same in each iteration, the contents of the second half were updated each time, to reflect recent developments in research. The book includes more than 350 exercises of varying difficulty, the majority of which have been tested in class. The book has appeared as volume 303 in Springer's Graduate Texts in Mathematics series, and is available here.

Publications

Books
Papers
Preprints
Other Manuscripts

Books

D	A Course in Real Algebraic Geometry. Positivity and Sums of Squares. Graduate Texts in Mathematics 303, Springer, 2024 (doi)
C	(with Manfred Knebusch) Real Algebra. A First Course. Translated and with Contributions by Thomas Unger. Universitext, Springer Nature, 2022 (doi)
B	Real and Étale Cohomology. Lecture Notes in Mathematics 1588, Springer, Berlin, 1994 (doi)
A	(with Manfred Knebusch) Einführung in die reelle Algebra. Vieweg, Wiesbaden, 1989 (doi)

Papers

71	(with Gennadiy Averkov) Convex hulls of monomial curves, and a sparse positivstellensatz. Mathematical Programming, published online 16 February 2024.
70	(with Igor Klep and Jurij Volcic) Globally trace-positive noncommutative polynomials and the unbounded tracial moment problem. Mathematische Annalen 387, 1403-1433 (2023). (pdf).
69	(with Tim Kobert) Spectrahedral representation of polar orbitopes. Manuscripta mathematica 169, 185-208 (2022). (pdf).
68	Extreme points of Gram spectrahedra of binary forms. Discrete and Computational Geometry 67, 1174-1190 (2022). (pdf).
67	Second-order cone representation for convex sets in the plane. SIAM J. Appl. Algebra Geometry 5, 114-139 (2021). (pdf).
66	(with Jose Capco) Two remarks on sums of squares with rational coefficients. Algebra, Logic and Number Theory, Banach Center Publ. 121, 25-36 (2020). (pdf)
65	(with Philipp Jell and Josephine Yu) Real tropicalization and analytification of semialgebraic sets. International Mathematics Research Notices 2022, 928-958. (pdf).
64	Spectrahedral shadows. SIAM J. Appl. Algebra Geometry 2, 26-44 (2018). (pdf).
63	Semidefinite representation for convex hulls of real algebraic curves. SIAM J. Appl. Algebra Geometry 2, 1-25 (2018). (pdf).
62	(with Colin Tan) A Positivstellensatz for forms on the positive orthant. Arch. Math. 109, 123--131 (2017). (doi)
61	(with Sebastian Wenzel) Polynomials nonnegative on the cylinder. In: Ordered Algebraic Structures and Related Topics, F. Broglia et al (eds.), Contemp. Math. 697, AMS, Providence, RI, 2017, pp. 291-300. (doi)
60	Sum of squares length of real forms. Math. Z. 286, 559-570 (2017). (doi))
59	(with Aaron Kunert) Extreme positive ternary sextics. Trans. Am. Math. Soc. 370, 3997--4013 (2018). (doi, preprint)
58	(with Daniel Plaumann) Toric completions and bounded functions on real algebraic varieties. J. London Math. Soc. (2) 94, 598-616 (2016).
57	Sums of squares of polynomials with rational coefficients. J. European Math. Soc. 18, 1495-1513 (2016).
56	(with Maria Charina, Mihai Putinar and Joachim Stöckler) An algebraic perspective on multivariate tight wavelet frames. II. Appl. Comput. Harmon. Anal. 39, 185-213 (2015).
55	(with David Harari and Tomás Szamuely) Weak approximation for tori over p-adic function fields. Int. Math. Res. Notices 10, 2751-2783 (2015).
54	(with Mihai Putinar) Quillen property of real algebraic varieties. Münster J. Math. 7, 671-696 (2014).
53	(with Maria Charina, Mihai Putinar and Joachim Stöckler) An algebraic perspective on multivariate tight wavelet frames. Constructive Approximation 38, 253-276 (2013)
52	Appendix to: "Isomorphism criteria for Witt rings of real fields" by N. Grenier-Boley and D. Hoffmann. Forum Math. 25, 15-17 (2013)
51	(with Mihai Putinar) Quillen property for real algebraic varieties. Illinois J. Math. 56, 213-220 (2012)
50	(with Albrecht Pfister) An elementary proof of Hilbert's theorem on ternary quartics. J. Algebra 371, 1-25 (2012)
49	(with Daniel Plaumann) The ring of bounded polynomials on a semi-algebraic set. Trans. Am. Math. Soc. 364, 4663-4682 (2012)
48	A Positivstellensatz for projective real varieties. Manuscr. math. 138, 73-88 (2012)
47	(with Sabine Burgdorf and Markus Schweighofer) Pure states, nonnegative polynomials, and sums of squares. Comm. Math. Helvetici 87, 113-140 (2012)
46	Convex hulls of curves of genus one. Adv. Math. 228, 2606-2622 (2011)
45	(with Mihai Putinar) Sums of hermitian squares on pseudoconvex boundaries. Math. Res. Lett. 17, 1047-1053 (2010)
44	The u-invariant of one-dimensional function fields over real power series fields. Arch. Math. 93, 245-251 (2009)
43	Weighted sums of squares in local rings and their completions, II. Math. Z. 266, 21-42 (2010)
42	Weighted sums of squares in local rings and their completions, I. Math. Z. 266, 1-19 (2010)
41	Hilbert's theorem on positive ternary quartics: A refined analysis. J. Algebraic Geometry 19, 285-333 (2010)
40	Positivity and sums of squares: A guide to recent results. In: Emerging Applications of Algebraic Geometry (M. Putinar, S. Sullivant, eds.), IMA Volumes Math. Appl. 149, Springer, 2009, pp. 271-324.
39	(with Jaka Cimprič and Salma Kuhlmann) Sums of squares and moment problems in equivariant situations. Trans. Am. Math. Soc. 361, 735-765 (2009)
38	(with Mihai Putinar) Multivariate moment problems: Geometry and indeterminateness. Ann. Scuola Norm. Sup. Pisa (5) 5, 137-157 (2006)
37	(with José F. Fernando and Jesús M. Ruiz) Sums of squares of linear forms. Math. Res. Lett. 13, 947-956 (2006)
36	Sums of squares on real algebraic surfaces. Manuscr. math. 119, 395-410 (2006)
35	Non-existence of degree bounds for weighted sums of squares representations. J. Complexity 21, 823-844 (2005)
34	Distinguished representations of non-negative polynomials. J. Algebra 289, 558-573 (2005)
33	(with Vicki Powers, Bruce Reznick and Frank Sottile) A new approach to Hilbert's theorem on ternary quartics. C. R. Acad. Sci. Paris, Ser. I 339, 617-620 (2004)
32	(with José F. Fernando and Jesús M. Ruiz) Sums of squares in real rings. Trans. Am. Math. Soc. 356, 2663-2684 (2004)
31	Sums of squares on real algebraic curves. Math. Z. 245, 725-760 (2003)
30	(with Joost van Hamel) Cohomology of tori over p-adic curves. Math. Ann. 326, 155-183 (2003)
29	(with David Lewis and Thomas Unger) A weak Hasse principle for central simple algebras with an involution. Doc. Math. Extra Volume, Proc. Conf. Quadratic Forms and Related Topics, Baton Rouge, La., 2001, 241-251 (2001)
28	On sums of squares in local rings. J. reine angew. Math. 540, 205-227 (2001)
27	(with Vicki Powers) The moment problem for non-compact semialgebraic sets. Adv. Geom. 1, 71-88 (2001)
26	Sums of squares of regular functions on real algebraic varieties. Trans. Am. Math. Soc. 352, 1039-1069 (1999)
25	The structure of some virtually free pro-p groups. Proc. Am. Math. Soc. 127, 695-700 (1999)
24	(with Yuval Z. Flicker and R. Sujatha) Grothendieck's theorem on non-abelian H² and local-global principles. J. Am. Math. Soc. 11, 731-750 (1998)
23	Appendix to: "A cohomological interpretation of the Grothendieck-Teichmüller group" by P. Lochak and L. Schneps. Invent. math. 127, 597-600 (1997)
22	Remark on the paper "Zerlegung reeller algebraischer Funktionen in Quadrate. Schiefkörper über reellem Funktionenkörper'' by E. Witt. In: Ernst Witt, Collected Papers --- Gesammelte Abhandlungen (I. Kersten, ed.), Springer, Berlin, 1998, p. 89
21	Classification of hermitian forms and semisimple groups over fields of virtual cohomological dimension one. Manuscr. math. 89, 373-394 (1996)
20	Hasse principles and approximation theorems for homogeneous spaces over fields of virtual cohomological dimension one. Invent. math. 125, 307-365 (1996)
19	Farrell cohomology and Brown theorems for profinite groups. Manuscr. math. 91, 247-281 (1996)
18	(with Jean-Louis Colliot-Thélène) Zero-cycles and cohomology on real algebraic varieties. Topology 35, 533-559 (1996)
17	Purity theorems for real spectra and applications. In: Real Analytic and Algebraic Geometry, Proc. Conf. Trento 1992, F. Broglia, M. Galbiati, A. Tognoli (eds.), de Gruyter, Berlin 1995, pp. 229-250
16	Quasi-augmented simplicial spaces, with an application to cohomological dimension. J. Pure Applied Algebra 81, 293-311 (1992)
15	Real algebra and its applications to geometry in the last ten years: Some major developments and results. In: Real Algebraic Geometry, Proc. Conf. Rennes 1991, M. Coste, L. Mahé, M.-F. Roy (eds.), Lect. Notes Math. 1524, Springer, Berlin, 1992, pp. 75-96
14	Some remarks on orderings under finite field extensions. Pacific J. Math. 152, 175-185 (1992)
13	Spaces of orderings of fields under finite extensions. Manuscr. math. 72, 27-47 (1991)
12	Stability index of real varieties. Invent. math. 97, 467-483 (1989)
11	(with Roland Huber) A relative notion of local completeness in semialgebraic geometry. Arch. Math. 53, 571-584 (1989)
10	Quotients of semi-algebraic spaces. Math. Z. 201, 249-271 (1989)
9	Riemannian almost-product structures with maximal mobility. Geom. Dedicata 24, 109-122 (1987)
8	Intersections of maximal subalgebras in Lie algebras. J. Algebra 105, 268-270 (1987)
7	Algebraic subgroup lattices of topological groups. Algebra Universalis 22, 235-243 (1986)
6	Topologies on the subgroup lattice of a compact group. Topology Appl. 23, 183-191 (1986)
5	Zusammenhängende Untergruppen von pro-Lieschen Gruppen. Geom. Dedicata 21, 231-248 (1986)
4	Quasi-normal subgroups of algebraic groups. Arch. Math. 45, 8-11 (1985)
3	Topologisch quasinormale Untergruppen zusammenhängender lokalkompakter Gruppen. Monatsh. Math. 98, 75-81 (1984)
2	Topologisch linksengelsche Elemente. Manuscr. math. 49, 243-266 (1985)
1	Topologisch rechtsengelsche Elemente in zusammenhängenden lokalkompakten Gruppen. Arch. Math. 43, 117-120 (1984)

Preprints

•	Convex hulls of curves in n-space (22 pp, October 2024) (pdf).
•	(with Albrecht Pfister) An elementary proof of Hilbert's theorem on ternary quartics: Some complements (6 pp, October 2024) (pdf).
•	(with Konrad Schmüdgen) Moment property and positivity for some algebras of fractions (14 pp, August 2024) (pdf).
•	An observation on positive definite forms. (2 pp, February 2016) (pdf).

Other Manuscripts

•	A remark on descending sums of squares representations (2 pp, 16 July 2009) (pdf)
•	In how many ways is a quartic a conic of conics? Data for irreducible quartics over C and for nonnegative absolutely irreducible quartics over R. (6 pp, 4 November 2007) (pdf)
•	A short remark on a theorem by Becker and Gondard (2 pp, 15 April 2004) (pdf)
•	Correction to [27] (pdf)
•	Correction to [26] (pdf)

A Course in Real Algebraic Geometry: Positivity and Sums of Squares