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)
TL;DR
This paper investigates the universality of generalized quadratic forms over real quadratic fields, establishing that such forms must contain universal quadratic subforms under certain conditions, and demonstrates the necessity of these conditions with an example.
Contribution
It proves that generalized quadratic forms over real quadratic fields are only universal if they include a universal quadratic subform, highlighting the importance of positive-definiteness.
Findings
Universal forms must contain a universal quadratic subform
Positive-definiteness condition is necessary for universality
Constructed example shows the necessity of the condition
Abstract
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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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Analytic Number Theory Research · Advanced Differential Equations and Dynamical Systems