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Ondrej Chwiedziuk,
Matej Dolezalek,
Emma Pechouckova,
Zdenek Pezlar,
Om Prakash,
Giuliano Romeo,
Anna Ruzickova, and
Mikulas Zindulka
Representing rational integers by generalized quadratic forms over quadratic fields
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| Published: |
May 3, 2026. |
| Keywords: |
Generalized quadratic form, universal quadratic form, Hermitian form, real quadratic field. |
| Subject [2010]: |
11E12, 11E16, 11E20, 11E25, 11E39, 11R11, 11R80. |
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Abstract
We investigate generalized quadratic forms with values in the set of rational integers over quadratic fields. We characterize the real quadratic fields which admit a positive definite binary generalized form of this type representing every positive integer. We also show that there are only finitely many such fields where a ternary generalized form with these properties exists.
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| Acknowledgements
We acknowledge support by Czech Science Foundation (GACR) grant 26-20514S, and Charles University programmes PRIMUS/24/SCI/010 and UNCE/24/SCI/022, and GAUK projects 134824 and 236225.
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| Author information
Ondrej Chwiedziuk
Charles University
Faculty of Mathematics and Physics, Department of Algebra
Sokolovska 83, 186 75 Praha 8, Czech Republic
ondrachwiedziuk@gmail.com
Matej Dolezalek
Charles University
Faculty of Mathematics and Physics, Department of Algebra
Sokolovska 83, 186 75 Praha 8, Czech Republic
matej@gimli.ms.mff.cuni.cz
Emma Pechouckova
Charles University
Faculty of Mathematics and Physics, Department of Algebra
Sokolovska 83, 186 75 Praha 8, Czech Republic
emma.pechouckova@gmail.com
Zdenek Pezlar
Charles University
Faculty of Mathematics and Physics, Department of Algebra
Sokolovska 83, 186 75 Praha 8, Czech Republic
zdendapezlar@seznam.cz
Om Prakash
Charles University
Faculty of Mathematics and Physics, Department of Algebra
Sokolovska 83, 186 75 Praha 8, Czech Republic
omprakash@ksom.res.in
Giuliano Romeo
Department of Mathematical Sciences "Giuseppe Luigi Lagrange"
Politecnico di Torino, Turin, Italy
giuliano.romeo@polito.it
Anna Ruzickova
Charles University
Faculty of Mathematics and Physics, Department of Algebra
Sokolovska 83, 186 75 Praha 8, Czech Republic
anna.ruzickova.13@gmail.com
Mikulas Zindulka
Charles University
Faculty of Mathematics and Physics, Department of Algebra
Sokolovska 83, 186 75 Praha 8, Czech Republic
mikulas.zindulka@matfyz.cuni.cz
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