Implicitization, Residual Intersections, and Differential Methods in Commutative Algebra

该奖项资助交换代数的研究，即对若干未知数多项式方程组的研究。为此，我们把一个方程组的所有解的集合看作一个几何对象，并研究在这个对象上定义的函数。相反的观点导致了隐化问题，这是研究者计划研究的问题：给定一个几何对象，如曲线或曲面，人们希望构建一个多项式方程系统，将几何对象作为其解集。一旦知道了多项式方程组，它就变得容易多了，例如，确定一个特定的点是否在表面上，或者表面在其中一个点上是否光滑。隐化问题是困难的，需要纯数学的先进技术，但即使在特殊情况下，它的解决方案也有许多应用，例如在计算机辅助设计、机器人和其他工程领域。本项目涉及交换代数中与代数、解析几何和消元理论有密切联系的几个主题。它们包括Rees代数和有理映射的隐化问题、等奇性理论、平面叶化的庞加莱问题和剩余交集理论。确定有理图和有理图的隐式方程是消去理论中的一个经典但开放的问题，它相当于找到定义Rees代数的理想。在此之前，PI和他的合作者解决了co维三维Gorenstein理想的Rees代数的这个问题，在附加的假设下，理想的合矩阵的条目产生一个完全相交。现在PI打算取消这个关键的假设。PI还计划研究更一般理想的Rees代数，目的是至少获得隐式方程的定性陈述和边界。等奇异性理论的一个目标是设计出一组解析空间拓扑平凡的精确数值准则。一个重要的中间步骤是模的积分相关性的数值表征。PI打算使用由交集理论启发的多重性概念来证明这样的特征。庞加莱提出了一个问题，即如何确定复平面上的一个奇异代数叶形是否具有作为叶形的代数曲线。在最近的时代，这个问题经常被视为一个关于向量场的不变量与曲线的不变量或向量场留下的不变量之间关系的问题。PI将利用他之前在代数微分和Castelnuovo-Mumford正则方面的专业知识来研究这个问题。残馀交集的概念，是连接或联络的一种推广，在交集理论和里斯代数的研究中是普遍存在的。其中最重要的是剩余交叉点的Cohen-Macaulayness和对偶性。基于部分结果和实验证据，David Eisenbud和PI观察到，出乎意料的是，许多残差相交，即使它们不是Cohen-Macaulay，也承认自对偶的秩1的最大Cohen-Macaulay模。PI和他的合作者打算证明这种不寻常的现象。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award funds research in Commutative Algebra, the study of systems of polynomial equations in several unknowns. To this end, one considers the collection of all solutions of a system of equations as a geometric object and investigates the functions defined on this object. Reversing the perspective leads to the implicitization problem, which the investigator plans to work on: Given a geometric object, such as a curve or a surface, one wishes to construct a system of polynomial equations that has the geometric object as its solution set. Once the system of polynomial equations is known, it becomes much easier, for instance, to decide whether a specific point lies on the surface or whether the surface is smooth at one of its points. The implicitization problem is difficult and requires advanced techniques from pure mathematics, but its solution even in particular cases has numerous applications, for instance in computer aided design, robotics, and other areas of engineering. This project addresses several topics in Commutative Algebra that have close connections with Algebraic and Analytic Geometry and with Elimination Theory. They include the implicitization problem for Rees algebras and rational maps, equisingularity theory, the Poincare problem for plane foliations, and residual intersection theory. Determining the implicit equations of graphs and images of rational maps is a classical, but open problem in elimination theory, which amounts to finding defining ideals of Rees algebras. Previously, the PI and his collaborators solved this problem for Rees algebras of codimension three Gorenstein ideals, under the additional assumption that the entries of a syzygy matrix of the ideal generate a complete intersection. Now the PI intends to remove this crucial hypothesis. The PI also plans to investigate Rees algebras of more general ideals, with the aim to obtain at least qualitative statements and bounds for the implicit equations. A goal in equisingularity theory is to devise fiberwise numerical criteria for when a family of analytic spaces is topologically trivial. An important intermediate step are numerical characterizations of integral dependence of modules. The PI intends to prove such a characterization using a notion of multiplicity that is inspired by intersection theory. Poincare had asked how to decide whether a singular algebraic foliation of the complex plane has an algebraic curve as a leaf. In more recent times, this question has often been treated as a problem about relating invariants of a vector field to invariants of curves or varieties that are left invariant by the vector field. The PI will investigate this problem, using his expertise from prior work on algebraic differentials and Castelnuovo-Mumford regularity. The notion of residual intersection, a generalization of linkage or liaison, is ubiquitous and appears naturally in intersection theory and in the study of Rees algebras, for instance. Of central importance are the Cohen-Macaulayness and duality properties of residual intersections. Based on partial results and experimental evidence, David Eisenbud and the PI have observed that, unexpectedly, many residual intersections, even when they fail to be Cohen-Macaulay, admit maximal Cohen-Macaulay modules of rank one that are self-dual. The PI and his collaborators intend to give a proof of this unusual phenomenon.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.