No proper generalized quadratic forms are universal over quadratic fields

Anna R\r{u}\v{z}i\v{c}kov\'a; Emma P\v{e}chou\v{c}kov\'a; Mat\v{e}j Dole\v{z}\'alek; Mikul\'a\v{s} Zindulka (Charles University); Om Prakash; Ond\v{r}ej Chwiedziuk; Simona Hlavinkov\'a; Zden\v{e}k Pezlar

arxiv: 2409.07941 · v1 · pith:G5LCDSJ3new · submitted 2024-09-12 · 🧮 math.NT

No proper generalized quadratic forms are universal over quadratic fields

Ond\v{r}ej Chwiedziuk , Mat\v{e}j Dole\v{z}\'alek , Simona Hlavinkov\'a , Emma P\v{e}chou\v{c}kov\'a , Zden\v{e}k Pezlar , Om Prakash , Anna R\r{u}\v{z}i\v{c}kov\'a , Mikul\'a\v{s} Zindulka (Charles University) This is my paper

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keywords quadraticgeneralizeduniversalconditionfieldsformspositive-definitenessconsider

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We consider generalized quadratic forms over real quadratic number fields and prove, under a natural positive-definiteness condition, that a generalized quadratic form can only be universal if it contains a quadratic subform that is universal. We also construct an example illustrating that the positive-definiteness condition is necessary.

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No proper generalized quadratic forms are universal over quadratic fields

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