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Two remarks on sums of squares with rational coefficients

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2020

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Capco, Jose

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https://doi.org/10.4064/bc121-2
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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, 2020, pp. 25-36. Banach Center Publications. 121. ISBN 9788386806478. Available under: doi: 10.4064/bc121-2

Zusammenfassung

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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510 Mathematik

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5th Joint Conferences on Algebra, Logic and Number Theory, 24. Juni 2018 - 29. Juni 2018, Będlewo
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ISO 690SCHEIDERER, Claus, Jose CAPCO, 2020. Two remarks on sums of squares with rational coefficients. 5th Joint Conferences on Algebra, Logic and Number Theory. Będlewo, 24. Juni 2018 - 29. Juni 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, 2020, pp. 25-36. Banach Center Publications. 121. ISBN 9788386806478. Available under: doi: 10.4064/bc121-2
BibTex
@inproceedings{Scheiderer2020remar-53177,
  year={2020},
  doi={10.4064/bc121-2},
  title={Two remarks on sums of squares with rational coefficients},
  number={121},
  isbn={9788386806478},
  publisher={Institute of Mathematics, Polish Academy of Sciences},
  address={Warschau},
  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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