Asymptotic growth of symbolic powers, mixed multiplicities, and convex bodies

这个项目属于交换代数，交换代数是数学的一个领域，研究多变量多项式方程的性质。科学和工程领域的各种现象都可以用多项式方程系统来建模，这导致了与应用数学、生物学、计算机科学和数学物理等其他领域的自然互动。对于多项式方程的系统，人们研究它的解的集合，称为变种，以及它们在高维空间中描述的相应形状。随着方程和变量的数量越来越多，理解这些形状的一个有用方法是研究在它们上面定义的代数函数。使用这种方法，许多属性，如平滑度和变化度可以转换成代数项。其中一种策略是通过分析称为理想的代数对象序列的渐近行为来研究变量的性质。该项目对这一研究方向做出了贡献，其问题源于两个方向，涉及对理想的这些性质的研究。PI还将参与指导学生和组织学术活动，重点是使研究生和数学领域代表性不足的群体受益。理想的符号能力编码了理想及其定义的多样性的重要代数和几何信息。PI将研究符号幂的同调不变量的渐近行为，如生成器的数量、Castelnuovo-Mumford正则性和射影维数。对于生成数序列，PI打算研究它是否总是具有多项式复杂度，建立PI和他的合作者之前的结果。这个方向的一般结果将对除数的算术秩、Frobenius复杂度和Kodaira维数产生重要影响。对于另外两个序列，规则性序列和射影维数序列，PI将关注具有正特征的正则环。在这样的环中，定义了一类新的理想，对于这些理想，先前已经证明了这些环存在极限；本课程包括几种类型的行列式理想，以及无平方单项式理想。PI将对更一般的理想类别研究这些极限的存在性。复数与凸体的关系是交换代数、代数几何和组合学相互作用中的一个重要研究课题。近年来，由于引入了牛顿-奥肯科夫体和理想过滤的混合多样性，这一研究领域获得了很大的活跃度。PI和他的合作者打算找到多重代数的混合多重不消失的条件。PI还将研究不一定是零维理想过滤的混合多重度的概念，以及这些新多重度与某些牛顿-奥库科夫体的混合体积的关系。该项目由数学科学部和促进竞争性研究的既定计划（EPSCoR）共同资助。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 project is in commutative algebra which is the area of mathematics that studies properties of polynomial equations in several variables. Various phenomena across science and engineering can be modeled by systems of polynomial equations, which leads to natural interactions with other fields such as applied mathematics, biology, computer science, and mathematical physics. For systems of polynomial equations, one studies its sets of solutions, called varieties, and the corresponding shapes they describe in high-dimensional spaces. As the number of equations and variables grow larger, one useful approach to understand these shapes is to study the algebraic functions defined on them. Using this approach, many properties such as smoothness and degrees of varieties can be translated into algebraic terms. One such strategy is to study properties of varieties by analyzing the asymptotic behavior of sequences of algebraic objects called ideals. This project contributes to this line of research with problems originating from two directions that involve the study of these properties of ideals. The PI will also be involved in mentoring students and in organizing academic events, focusing on benefiting graduate students and underrepresented groups in mathematics.The symbolic powers of an ideal encode important algebraic and geometric information of the ideal and the variety it defines. The PI will investigate the asymptotic behavior of homological invariants of symbolic powers such as, the number of generators, the Castelnuovo-Mumford regularity, and the projective dimension. For the sequence of numbers of generators, the PI intends to investigate if it always has polynomial complexity, building of previous results by the PI and his collaborators. A general result in this direction would have important consequences on the arithmetic rank, Frobenius complexity, and Kodaira dimension of divisors. For the other two sequences, regularities and projective dimensions, the PI will focus on regular rings of positive characteristic. In such rings, a new class of ideals is defined for which it has been previously shown the limit of these sequences exist; this class includes several types of determinantal ideals, as well as the square-free monomial ideals. The PI will investigate the existence of these limits for more general classes of ideals. The relation between multiplicities and convex bodies is an important research topic lying in the interaction of Commutative Algebra, Algebraic Geometry, and Combinatorics. In recent years, this line of research has gained much activity due to the introduction of Newton-Okounkov bodies and mixed multiplicities of filtrations of ideals. The PI and his collaborators intend to find conditions for the non-vanishing of mixed multiplicities of multigraded algebras. The PI will also investigate a notion of mixed multiplicities for filtrations of not necessarily zero dimensional ideals and the relation of these new multiplicities with the mixed volumes of certain Newton-Okounkov bodies.This project is jointly funded by the Division of Mathematical Sciences and the Established Program to Stimulate Competitive Research (EPSCoR).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.