Type of Publication: | Contribution to a conference collection |
Publication status: | Published |
Author: | Scheiderer, Claus; Capco, Jose |
Year of publication: | 2020 |
Conference: | 5th Joint Conferences on Algebra, Logic and Number Theory, Jun 24, 2018 - Jun 29, 2018, Będlewo |
Published in: | Proceedings of the 5th Joint Conferences on Algebra, Logic and Number Theory / Gładki, Paweł; Koenigsmann, Jochen; Koprowski, Przemysław et al. (ed.). - Warschau : Institute of Mathematics, Polish Academy of Sciences, 2020. - (Banach Center Publications ; 121). - pp. 25-36. - ISBN 9788386806478 |
DOI (citable link): | https://dx.doi.org/10.4064/bc121-2 |
Summary: |
There exist homogeneous polynomials f with Q-coefficients that are sums of squares over R but not over Q. The only systematic construction of such polynomials that is known so far uses as its key ingredient totally imaginary number fields K/Q with specific Galois-theoretic properties. We first show that one may relax these properties considerably without losing the conclusion, and that this relaxation is sharp at least in a weak sense. In the second part we discuss the open question whether any f as above necessarily has a (non-trivial) real zero. In the minimal open cases (3,6) and (4,4), we prove that all examples without a real zero are contained in a thin subset of the boundary of the sum of squares cone.
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Subject (DDC): | 510 Mathematics |
Bibliography of Konstanz: | Yes |
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SCHEIDERER, Claus, Jose CAPCO, 2020. Two remarks on sums of squares with rational coefficients. 5th Joint Conferences on Algebra, Logic and Number Theory. Będlewo, Jun 24, 2018 - Jun 29, 2018. In: GŁADKI, Paweł, ed., Jochen KOENIGSMANN, ed., Przemysław KOPROWSKI, ed. and others. Proceedings of the 5th Joint Conferences on Algebra, Logic and Number Theory. Warschau:Institute of Mathematics, Polish Academy of Sciences, pp. 25-36. ISBN 9788386806478. Available under: doi: 10.4064/bc121-2
@inproceedings{Scheiderer2020remar-53177, title={Two remarks on sums of squares with rational coefficients}, year={2020}, doi={10.4064/bc121-2}, number={121}, isbn={9788386806478}, address={Warschau}, publisher={Institute of Mathematics, Polish Academy of Sciences}, series={Banach Center Publications}, booktitle={Proceedings of the 5th Joint Conferences on Algebra, Logic and Number Theory}, pages={25--36}, editor={Gładki, Paweł and Koenigsmann, Jochen and Koprowski, Przemysław}, author={Scheiderer, Claus and Capco, Jose} }
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