论文标题
我们可以通过表示其所有子形式来恢复整体二次形式吗？
Can we recover an integral quadratic form by representing all its subforms?
论文作者
论文摘要
令$ \ mathfrak o $为完全真实数字字段的整数环。如果$ f $是$ \ mathfrak o $的二次形式，而$ g $是$ \ m athfrak o $的另一种二次表格，代表$ f $的所有适当子形式，$ g $表示$ g $表示$ f $？我们表明，如果$ g $是不确定的，那么$ g $确实代表$ f $。但是，当$ f $是正定且难以解决的时，就会存在$ g $，代表$ f $的所有适当子形式，而不是$ f $本身。一路走来，我们给出了$ \ mathfrak o $的积极确定的二次二次形式的新特征，以及对二次形式的有限形式理论的数字概括，而二次形式的形式超过了$ \ mathbb z $，这些形式在$ \ mathbb z $上所主张，这些表格当然，鉴于任何无限级的$ \ mathscr $ \ mathscr $ \ maths $ hecteral $ \ natile $ \ natile $ \ natire的$ \ \ \ \ \ \ \ mather的均值，存在有限的子集$ \ MATHSCR S_0 $ $ \ MATHSCR S $的属性，该属性是$ \ Mathfrak O $上的正确定二次形式，代表$ \ Mathscr S $中的所有类，仅当它代表$ \ Mathscr s_0 $中的所有类时，它才代表$ \ mathscr s $。
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 $\mathscr S_0$.
