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Rational points and lines on cubic hypersurfaces

dc.contributor.advisorBrüdern, Jörg Prof. Dr.
dc.contributor.authorBernert, Christian
dc.titleRational points and lines on cubic hypersurfacesde
dc.contributor.refereeBrüdern, Jörg Prof. Dr.
dc.description.abstractengWe study the solubility of cubic diophantine equations. In the first chapter, we discuss the convergence of the singular series of a cubic form, which is a central object in the study of the solutions of the associated cubic equation. We establish new bounds for the relevant exponential sums (Gauß sums) and use these to establish the absolute convergence of the singular series in the critical case of 10 variables. In the second chapter, which is joint work with Leonhard Hochfilzer, we use the Hardy-Littlewood Circle Method to prove the existence of solutions to homogeneous cubic equations in at least 14 variables over imaginary quadratic number fields. As an application, we deduce the existence of a rational line on a cubic hypersurface in at least 33 variables, improving on previous work by Wooley. In the final chapter, we study inhomogeneous cubic equations and establish the existence of solutions in new cases. Moreover, we provide a method to bound the smallest solution of such an equation, which also has interesting applications in the homogeneous
dc.contributor.coRefereeSchindler, Damaris Prof. Dr.
dc.subject.engExponential sumsde
dc.subject.engAnalytic number theoryde
dc.subject.engHardy-Littlewood Circle Methodde
dc.subject.engCubic formsde
dc.subject.engRational pointsde
dc.subject.engDiophantine equationsde
dc.subject.engNumber fieldsde
dc.affiliation.instituteFakultät für Mathematik und Informatikde
dc.subject.gokfullMathematics (PPN61756535X)de
dc.notes.confirmationsentConfirmation sent 2023-10-02T19:45:01de

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