Foundations of Asymptotic Differential Algebra

渐近微分代数基础

• 批准号: 0556197
• 负责人: Matthias Aschenbrenner
• 金额: $ 15.83万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0556197&HistoricalAwards=false
• 关键词: Foundations; Asymptotic; Differential; Algebra

Aschenbrenner proposes to contribute to the fields of model theory and algebra. He proposes to continue his collaborative efforts to create a synthesis between the fields of real algebraic geometry and differential algebra, in the form of asymptotic differential algebra. This will lead to a deeper understanding of the model-theoretic and algebraic properties of Hardy fields and fields of transseries, and the relationship between them, and will open up new perspectives in the asymptotic theory of algebraic differential equations. He also proposes to continue his investigations into non-standard methods and degree bounds in commutative algebra. For example, Aschenbrenner proposes to increase our knowledge about Grobner bases for ideals in the ring of polynomials with integral coefficients.Mathematical logic, although it originated in the late 19th century with philosophical investigations into the foundations of mathematics, has in recent decades found many applications in other parts of mathematics, in computer science, and even in engineering (quantifier elimination as it relates to robotics). This project deals mainly with pushing the applicability of the methods of mathematical logic into as of yet unexplored territory, namely, asymptotic analysis. Our hope is that these investigations will lead to a deeper understanding of the behavior of solutions to differential equations. Another part of the proposed project involves the construction and analysis of algorithms for algebraic problems. Finding degree bounds as suggested in this project is of central importance for applications of computer algebra in science and engineering.

Aschenbrenner建议有助于模型理论和代数领域。他建议继续他的合作努力，创造一个综合领域之间的真实的代数几何和微分代数，在形式的渐近微分代数。 这将导致更深入地了解哈代场和transseries场的模型理论和代数性质，以及它们之间的关系，并将开辟新的视角在渐近理论的代数微分方程。他还建议继续他的调查非标准的方法和程度界限交换代数。例如，Aschenbrenner建议增加我们对整数系数多项式环中理想的Grobner基的知识。数理逻辑虽然起源于19世纪后期对数学基础的哲学研究，但在最近几十年里在数学的其他部分，计算机科学，甚至工程学（与机器人技术有关的量词消除）中找到了许多应用。这个项目主要涉及将数理逻辑方法的适用性推进到尚未探索的领域，即渐近分析。我们的希望是，这些调查将导致更深入地了解微分方程的解决方案的行为。该项目的另一部分涉及代数问题算法的构建和分析。 找到本项目中建议的学位界限对于计算机代数在科学和工程中的应用至关重要。