微分差分代数的相交理论与高效消元算法

微分差分代数是研究微分差分方程的代数理论，具有可构造性与可机械化计算的特色，是数学机械化的重要研究领域。相交理论与消元算法是微分差分代数的核心研究内容，其发展面临诸多困难与挑战，如经典公开问题微分维数猜想与微分Jacobi界猜想尚未解决。近年来，申请人与合作者建立了常微分(差分)多项式系统的周形式与稀疏结式的理论及算法，在两个猜想的研究上取得了阶段性进展，但仍有很多问题亟待深入研究。本项目将以前期工作为基础，继续深入研究微分差分代数的相交理论与高效消元算法。具体研究内容包括：1)研究退化情形下的微分维数猜想与微分Jacobi界猜想；2)发展偏微分周形式理论与常微分Hilbert概型；3)发展偏微分稀疏结式理论与高效消元算法；4)研究偏微分-差分混合系统的有效Hilbert零点定理问题。以上都是微分差分代数的前沿问题，项目的顺利实施将会实质性推进微分差分代数的发展，拓宽数学机械化的适用范围。

Differential Algebra and difference algebra are branches of Mathematics aiming to study algebraic differential and difference equations based on the methods of commutative algebra and computational algebraic geometry. Due to their constructive and computational character, differential algebra and difference algebra naturally become important research fields of mathematics mechanization. Intersection theory and elimination theory are central parts in differential and difference algebra, and are still far from well-developed up to now. There are a lot of challenging unsolved problems in differential algebra, among which the differential dimension conjecture and the Jacobi bound conjecture are two classical open problems in differential elimination theory. Recently, in ordinary differential algebra, we established the theory of differential Chow forms, differential Chow varieties, and sparse differential resultants. Also, we made progress in the study of the above two conjectures. As remarked by the international peers, these are original and substantial contributions to the development of differential algebra. However, it is just the beginning of such study and a lot of more challenging new problems arise...Based on the known work, in this project, on the one hand, we will deeply explore the theory of differential Chow forms and sparse differential resultant in both ordinary and partial differential case, and on the other hand, we hope to make new progress in the study of the two conjectures. To be more specific, we will study the following problems: 1) Study the differential dimension conjecture with the method of differential specialization,and apply the model theory of differential valued fields to deal with the differential dimension conjecture, hoping to find sufficient conditions for the validity of these two conjectures; 2) Develop the theory of partial differential Chow forms and ordinary differential Hilbert scheme; 3) Study the theory of sparse differential resultant in the partial differential case,devise more efficient algorithms to compute sparse differential resultants; 4) Give upper bounds to solve the partial differential-difference version of the efficient Hilbert Nullstellensatz problem. These problems are frontiers of differential and difference algebra. The project will substantially promote the study of constructive differential and difference algebra, and enrich the theory and applications of mathematics mechanization.