UNIVERSAL GENERALISED QUADRATIC FORMS REDUCE TO THEIR QUADRATIC SUBFORMS

Abstract Chwiedziuk et al. [‘No proper generalized quadratic forms are universal over quadratic fields’, Ramanujan J. 69 (2026), Article no. 92] recently proved that over a real quadratic field, a totally positive definite universal generalised quadratic form must contain a universal quadratic subform and asked whether this fails in higher degree. We prove it never does. We establish a sharper statement, valid over every totally real field: if such a form represents a value using a nonzero proper variable, that value is bounded below in at least two Archimedean embeddings. Universality follows at once, since a target can be made small in d − 1 $d-1$ d minus 1 of the d $d$ d embeddings simultaneously, the unit rank of a totally real field of degree d $d$ d being d − 1 $d-1$ d minus 1 . The bound two is sharp. We analyse examples to explain the mechanisms governing the result in degrees 2 $2$ 2 , 3 $3$ 3 and 4 $4$ 4 .