- Titel
- Quadratic Determinantal Representations of Nonnegative Ternary Polynomials
- Zitierfähige Url:
- https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa2-1017043
- Erstveröffentlichung
- 2026
- Datum der Einreichung
- 10.09.2025
- Datum der Verteidigung
- 13.01.2026
- Abstract (EN)
- This thesis analyzes the existence of positive quadratic determinantal representations of nonnegative ternary polynomials. It is proven that the Robinson polynomial does not admit such a representation, which answers a previously open conjecture by [BˇS20] to the positive. The methods to prove this involve the normalization of curves and sheaf cohomology to reduce the problem to a finite number of computations, which were performed using the computer algebra system Macaulay2. As a positive result, it is next proven that every polynomial of degree four that defines a nonsingular curve has exactly four distinct positive quadratic representations. This result heavily relies on prior work on the combinatorial structure of nonsingular quartic curves, specifically their bitangent lines. An algorithm to compute these representations is described. The thesis concludes by an alternative characterization of the positive quadratic representations of a smooth quartic curve. This is achieved by interpreting the quartic as the ramification locus of the anticanonical morphism of a nonsingular del Pezzo surface of degree two.
- Freie Schlagwörter (DE)
- Positive Polynome, Determinanten-Darstellungen, del-Pezzo-Flächen, Reelle Algebraische Geoemetrie
- Freie Schlagwörter (EN)
- Positive Polynomials, Determinantal Representations, del Pezzo Surfaces, Real Algebraic Geometry
- Klassifikation (DDC)
- 510
- Klassifikation (RVK)
- SK 240
- SK 180
- GutachterIn
- Jun.Prof. Dr. Mario Kummer
- Prof. Grigoriy Blekherman
- BetreuerIn Hochschule / Universität
- Jun.Prof. Dr. Mario Kummer
- Prof. Dr. Andreas Thom
- Den akademischen Grad verleihende / prüfende Institution
- Technische Universität Dresden, Dresden
- Förder- / Projektangaben
- Deutsche Forschungsgemeinschaft Real algebraic geometry, convexity and topology
ID: 502861109 - Version / Begutachtungsstatus
- publizierte Version / Verlagsversion
- URN Qucosa
- urn:nbn:de:bsz:14-qucosa2-1017043
- Veröffentlichungsdatum Qucosa
- 21.01.2026
- Dokumenttyp
- Dissertation
- Sprache des Dokumentes
- Englisch
- Lizenz / Rechtehinweis
CC BY-NC 4.0- Inhaltsverzeichnis
Abstract Thesis Outline and Main Results I Preliminaries and Survey of Prior Work I.1 General Tools from Algebraic Geometry I.2 Determinantal Representations of Plane Curves II Ternary Polynomials Without Positive Quadratic Representations# II.1 General observations II.2 Resolution of Isolated Real Nodes II.3 An Example in Degree Four II.4 The Robinson Polynomial III Positive Quadratic Representations of Ternary Quartics III.1 General Observations III.2 Quartic Curves with Empty Real Part III.3 Double Covers and Degree Two del Pezzo Surfaces Conclusion and Outlook A Code A.1 The 'Quarez' Polynomial A.2 The Robinson Polynomial A.3 The Double Cover of a Quartic Curve Bibliography Eidesstattliche Erklärung