论文标题
张量限制定理对有限字段
A Tensor Restriction Theorem over Finite Fields
论文作者
论文摘要
限制是$ d $ - 道路张量的天然准排。在固定有限场上的张量时，我们建立了这一准阶的一个非凡的方面 - 即，它是一个良好的顺序：它不承认无限敌抗，也没有无限降低序列。这个结果让人联想到图形次要定理，对任意限制限制的张量属性$ x $有重要影响。例如，$ x $承认有限的许多禁止限制的特征，可以通过查看固定尺寸的子操作器来测试。我们的证明涉及对多项式通用表示形式的诱导，将张量限制定理的概括（例如固定程度的均质多项式）概括，并描述了任何限制性封闭特性的粗略结构。
Restriction is a natural quasi-order on $d$-way tensors. We establish a remarkable aspect of this quasi-order in the case of tensors over a fixed finite field -- namely, that it is a well-quasi-order: it admits no infinite antichains and no infinite strictly decreasing sequences. This result, reminiscent of the graph minor theorem, has important consequences for an arbitrary restriction-closed tensor property $X$. For instance, $X$ admits a characterisation by finitely many forbidden restrictions and can be tested by looking at subtensors of a fixed size. Our proof involves an induction over polynomial generic representations, establishes a generalisation of the tensor restriction theorem to other such representations (e.g. homogeneous polynomials of a fixed degree), and also describes the coarse structure of any restriction-closed property.
