I thank my colleagues... ... Karim J. Becher for giving me an interesting research question and f... more I thank my colleagues... ... Karim J. Becher for giving me an interesting research question and for his support and advice; ... Julia Hartmann for an inspiring course on the topic covered in [HH10] and [HHK09]; ... Kevin Hutchinson and his colleagues University College Dublin for their hospitality during a predoctoral research stay; ... David Leep for his encouragement and for inspiring discussions beyond the scope of this thesis; ... Claus Scheiderer for carefully reviewing my thesis and spotting non-neglectable mistakes before the final print; ... Jan Van Geel for his ideas and for being my go-to-guy for all geometry-related problems; ... Adrian Wadsworth for helping me find a more conceptual proof of a main result; .
A recently found local-global principle for quadratic forms over function fields of curves over a... more A recently found local-global principle for quadratic forms over function fields of curves over a complete discrete valued field is applied to the study of quadratic forms, sums of squares, and related field invariants.
Isotropy of quadratic forms (resp. principal homogeneous spaces) over function fields of p-adic c... more Isotropy of quadratic forms (resp. principal homogeneous spaces) over function fields of p-adic curves was characterized by Harbater, Hartmann and Krashen using field patching techniques: If isotropy holds in all patches then it also holds in the function field. Colliot-Thélène, Parimala and Suresh used this isotropy criterion to derive a local-global principle for quadratic forms that invokes the completions of the function field with respect to its discrete valuations. In this note, we present an intermediate result.
We consider real polynomials in finitely many variables. Let the variables consist of finitely ma... more We consider real polynomials in finitely many variables. Let the variables consist of finitely many blocks that are allowed to overlap in a certain way. Let the solution set of a finite system of polynomial inequalities be given, where each inequality involves only variables of one block. We investigate polynomials that are positive on such a set and sparse in the sense that each monomial involves only variables of one block. In particular, we derive a short and direct proof for Lasserre’s theorem on the existence of sums of squares certificates respecting the block structure. The motivation for the results can be found in the literature on numerical methods for global optimization of polynomials that exploit sparsity.
I thank my colleagues... ... Karim J. Becher for giving me an interesting research question and f... more I thank my colleagues... ... Karim J. Becher for giving me an interesting research question and for his support and advice; ... Julia Hartmann for an inspiring course on the topic covered in [HH10] and [HHK09]; ... Kevin Hutchinson and his colleagues University College Dublin for their hospitality during a predoctoral research stay; ... David Leep for his encouragement and for inspiring discussions beyond the scope of this thesis; ... Claus Scheiderer for carefully reviewing my thesis and spotting non-neglectable mistakes before the final print; ... Jan Van Geel for his ideas and for being my go-to-guy for all geometry-related problems; ... Adrian Wadsworth for helping me find a more conceptual proof of a main result; .
A recently found local-global principle for quadratic forms over function fields of curves over a... more A recently found local-global principle for quadratic forms over function fields of curves over a complete discrete valued field is applied to the study of quadratic forms, sums of squares, and related field invariants.
Isotropy of quadratic forms (resp. principal homogeneous spaces) over function fields of p-adic c... more Isotropy of quadratic forms (resp. principal homogeneous spaces) over function fields of p-adic curves was characterized by Harbater, Hartmann and Krashen using field patching techniques: If isotropy holds in all patches then it also holds in the function field. Colliot-Thélène, Parimala and Suresh used this isotropy criterion to derive a local-global principle for quadratic forms that invokes the completions of the function field with respect to its discrete valuations. In this note, we present an intermediate result.
We consider real polynomials in finitely many variables. Let the variables consist of finitely ma... more We consider real polynomials in finitely many variables. Let the variables consist of finitely many blocks that are allowed to overlap in a certain way. Let the solution set of a finite system of polynomial inequalities be given, where each inequality involves only variables of one block. We investigate polynomials that are positive on such a set and sparse in the sense that each monomial involves only variables of one block. In particular, we derive a short and direct proof for Lasserre’s theorem on the existence of sums of squares certificates respecting the block structure. The motivation for the results can be found in the literature on numerical methods for global optimization of polynomials that exploit sparsity.
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