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Can we recover an integral quadratic form by representing all its subforms?
Number Theory · Submitted 22 Jan 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim However, when $f$ is positive definite and indecomposable, then there exists a $g$ which represents all proper subforms of $f$ but not $f$ itself.
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The abstract, sentence by sentence
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 $\mathscr S_0$ 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 $\mathscr S_0$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $\mathfrak o$ be the ring of integers of a totally real number field. | background 1.00 | 0.05 | 0.01 | asserted 0.88 |
| s1 | 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 $f$? | background 0.94 | 0.05 | 0.04 | unclear 0.78 |
| s2 | We show that if $g$ is indefinite, then $g$ indeed represents $f$. | result 1.00 | 0.95 | 0.36 | asserted 1.00 |
| s3 | However, when $f$ is positive definite and indecomposable, then there exists a $g$ which represents all proper subforms of $f$ but not $f$ itself. | result 0.95 | 0.91 | 0.28 | asserted 1.00 |
| s4 | 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 $\mathscr S_0$ 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 $\mathscr S_0$. | contribution 0.89 | 0.91 | 0.08 | asserted 1.00 |
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The paper
Original abstract
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 $\mathscr S_0$ 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 $\mathscr S_0$.
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How the literature uses it
3 arXiv papers cite this in 3 sentences
Papers citing it · 1 sentence without a confident label
Cites it as background 2 papers
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For more information on quadratic forms having an isolation, one may see [ref] and [ref] .
Isolations of the sum of two squares from its proper subforms · Introduction · 1.00
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For some more relevant background and interesting results on quadratic lattices, see, e.g., [ref] .
Real quadratic fields with a universal quadratic form of given rank have density zero · Large ranks of universal forms · 1.00
What it cites
Cites as background 2 papers
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The number-field extension of this result has been sought after in some recent work on universal quadratic forms (see, for example, [ref] and [ref] ).
Number fields without universal quadratic forms of small rank exist in most degrees · The Finiteness Theorem · 0.91
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The number-field extension of this result has been sought after in some recent work on universal quadratic forms (see, for example, [ref] and [ref] ).
On Kitaoka's conjecture and lifting problem for universal quadratic forms · The Finiteness Theorem · 0.91
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