Relative rank and regularization
Relative rank and regularization
复制标题
相对排名和正则化
DOI：
--
复制
发表时间：
2021
期刊：
影响因子：
--
通讯作者：
T. Ziegler
中科院分区：
文献类型：
--
作者：
Amichai Lampert;T. Ziegler
摘要本文提出了一种新的秩的概念--与多项式过滤集合相关的秩.当过滤是平凡的，我们的相对排名符合施密特排名（也称为强度）。我们还介绍了相对偏差的概念。本文的主要结果是有限域上这两个量之间的关系（作为特例，我们得到了文[21]中结果的一个新证明）。这种关系使我们能够得到一个精确的估计点的数量的仿射品种所提供的一个集合的多项式是高的相对秩（引理3.2）。相对秩的关键优点是它允许执行有效的正则化过程，该过程在多项式的初始数量中是多项式的（具有施密特秩的正则化过程比塔指数差得多）。主要结果使我们能够取代施密特秩与相对秩在许多关键的应用组合，代数几何，代数。例如，我们证明任何多项式集合
$mathcal P=（P_i）_{i=1}^c$
度
$le d$
在代数闭特征域上的多项式环中
$>d$
包含在一个理想中
$mathcal I（{mathcal Q}）$
，由集合生成
${mathcal Q}$
次数多项式的
$le d$
它们形成规则序列，并且
${mathcal Q}$
大小为
$le A c^{A}$
得双曲余切值.
A=A（d）
它与变量的数量无关。
Abstract We introduce a new concept of rank – relative rank associated to a filtered collection of polynomials. When the filtration is trivial, our relative rank coincides with Schmidt rank (also called strength). We also introduce the notion of relative bias. The main result of the paper is a relation between these two quantities over finite fields (as a special case, we obtain a new proof of the results in [21]). This relation allows us to get an accurate estimate for the number of points on an affine variety given by a collection of polynomials which is of high relative rank (Lemma 3.2). The key advantage of relative rank is that it allows one to perform an efficient regularization procedure which is polynomial in the initial number of polynomials (the regularization process with Schmidt rank is far worse than tower exponential). The main result allows us to replace Schmidt rank with relative rank in many key applications in combinatorics, algebraic geometry, and algebra. For example, we prove that any collection of polynomials
$mathcal P=(P_i)_{i=1}^c$
of degrees
$le d$
in a polynomial ring over an algebraically closed field of characteristic
$>d$
is contained in an ideal
$mathcal I({mathcal Q})$
, generated by a collection
${mathcal Q}$
of polynomials of degrees
$le d$
which form a regular sequence, and
${mathcal Q}$
is of size
$le A c^{A}$
, where
$A=A(d)$
is independent of the number of variables.
影响因子：
3.9
作者：
Ananyan, Tigran;Hochster, Melvin
通讯作者：
Hochster, Melvin