2022/01/22 by Wai Kiu Chan, Byeong-Kweon Oh, Chan, Wai Kiu +1 · 4 citations
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2201.08957
openalex publication_date 2022/01/22 · openalex created_date 2022/04/03 · openalex updated_date 2026/08/01
Let \mathfrak o be the ring of integers of a totally real number field. If f is a quadratic form over \mathfrak o and g is another quadratic form over \mathfrak o which represents all proper subforms of f, does g represent f? We show that if g is indefinite, then g indeed represents f. However, when f is positive definite and indecomposable, then there exists a g which represents all proper subforms of f but not f itself. Along the way we give a new characterization of positive definite decomposable quadratic forms over \mathfrak o and a number-field generalization of the finiteness theorem of representations of quadratic forms by quadratic forms over \mathbb Z which asserts that given any infinite set \mathscr S of classes of positive definite integral quadratic forms over \mathfrak o of a fixed rank, there exists a finite subset \mathscr S0 of \mathscr S with the property that a positive definite quadratic form over \mathfrak o represents all classes in \mathscr S if and only if it represents all classes in \mathscr S0.