微分、差分周形式与稀疏结式的理论与高效算法

The Chow form and the sparse resultant are both basic concepts in algebraic geometry and also powerful tools in elimination theory. Given the fact that Chow forms and resultants play an important role in both theoretic and algorithmic aspects of algebraic geometry and have applications in many fields, it is worthwhile to develop the theory of Chow forms and resultants in both differential and difference algebraic geometry, and hope that they will play a similar role as their algebraic counterparts. Recently, we have initially established the theory of differential chow form and sparse differential and difference resultant, but this is only the beginning of the research and there are still many problems that can be further explored in both theoretic and algorithmic aspects. Also, difference Chow form is not developed either. Based on our recent work, this project mainly studies the unsolved problems in the theory of differential Chow form and sparse differential resultant, as well as developing the theory of difference Chow form. To be more precise, in theoretical aspects, we plan to prove the existence of differential Chow variety for differential cycles and develop the theory of difference Chow form; in computing aspects, devise efficient algorithms to compute both differential Chow form and sparse differential resultant, and give matrix formulas for sparse differential resultants; in applied aspects, explore the potential applications of differential Chow forms and sparse differential resultants, and we expect that some of the problems that can be solved efficiently in the algebraic case can also be solved in the differential case, such as the effective differential nullstellensatz problem. The project will substantially promote the study of constructive differential algebraic geometry.

代数周形式与稀疏结式是代数几何的基本概念，也是消去理论的强大工具，在很多领域有重要应用，很多具有最佳复杂度的算法都基于这两个工具。一个自然的想法是在微分代数几何与差分代数中发展相应的周形式与稀疏结式理论。目前，我们已初步建立了微分周形式与稀疏微分、稀疏差分结式理论，并给出了计算稀疏微分、差分结式的单指数时间算法，但是这仅仅是相关研究的开始，无论是理论研究还是算法设计都需要进一步丰富和完善，而差分周形式至今没有得到发展。本项目将在我们已有工作的基础上，进一步研究微分、差分周形式与稀疏结式的理论与高效算法。在理论上，我们计划证明微分周簇的存在性，发展差分周形式以及微分差分Toric簇理论等；在算法上，设计计算微分周形式与稀疏微分结式的矩阵表示与高效算法；在应用上，将研究成果应用到有效微分零点定理最优界问题以及涉及微分、差分方程的实际应用问题中。本项目的研究将实质性推进构造性微分代数几何的研究。

代数周形式与稀疏结式是代数几何的基本概念，也是消去理论的强大工具，在很多领域有重要应用，很多具有最佳复杂度的算法都基于这两个工具。一个自然的想法是在微分代数几何与差分代数中发展相应的周形式与稀疏结式理论。项目负责人的博士论文初步建立了微分周形式与稀疏微分结式理论，但是这仅仅是相关研究的开始，无论是理论研究还是算法设计都需要进一步丰富和完善，而差分周形式没有得到发展。.. 本项目以已有工作为基础，在微分、差分周形式与稀疏结式的理论与高效算法方面开展了深入系统的研究，取得了一系列重要的研究成果。在理论方面，我们证明了微分周簇的存在性，从而完整解决了微分周簇是否存在这一公开问题，填充了微分代数几何中模空间研究的空白；在差分代数中发展了差分周形式理论；系统建立了稀疏微分结式理论；证明了特殊情形的Jacobi界猜想。在算法方面，分别设计了计算稀疏微分结式与计算微分周形式的高效算法。以上成果被称为对微分代数几何与差分代数做出了实质性与原创性的贡献。本项目共发表文章 5 篇，其中在计算理论顶尖期刊 FoCM 发表长论文(67页) 1 篇，伦敦数学会期刊JLMS 1 篇，Journal of Algebra 1 篇，Advances in Applied Mathematics 1 篇，以及会议(EI)论文1篇。