微分周形式与稀疏微分结式

Differential Chow form and sparse differential resultant

代数周(Chow)形式和代数结式是代数几何的基本概念,同时还是消去理论的强大工具.一个自然的想法是在微分代数几何中发展相应的周形式和结式理论.但是由于微分结构的复杂性,在本文的研究工作之前,微分结式只有部分结果,而微分周形式与稀疏微分结式理论一直没有得到发展.本文的主要结果包括:第一,发展一般(generic)情形的微分相交理论,作为应用,证明一般情形的微分维数猜想.第二,初步建立微分周形式理论.对不可约微分代数簇定义微分周形式并证明其基本性质,特别地,给出微分周形式的Poisson分解公式,引入微分代数簇的主微分次数这一不变量并证明一类微分代数闭链的周簇和周坐标的存在性.作为应用,首次严格定义微分结式,证明其基本性质.第三,初步建立稀疏微分结式理论.引入Laurent微分本性系统的概念,定义稀疏微分结式,证明其基本性质,特别地,引入微分环面簇的概念,给出稀疏微分结式阶数和次数界的估计,并基于此给出计算稀疏微分结式的单指数时间算法.

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 they play an important role in both theoretic and algorithmic aspects of algebraic geometry, it is worthwhile to develop the theory of Chow forms and resultants in differential algebraic geometry. But due to the the complicated structure of differential polynomials, the theory of resultants in differential case is not fully explored and the differential Chow form as well as the sparse differential resultant is not studied before. The main results in this work include the following three parts： Firstly, an intersection theory for generic differential polynomials is presented. As a consequence, the dimension conjecture for generic differential polynomials is proved. Secondly, the Chow form for an irreducible differential variety is defined and most of the properties of the Chow form in the algebraic case are established for its differential counterpart. In particular, a Possion-type product formula for differential Chow forms is given, the concept of the leading differential degree is introduced, and the existence of the differential Chow variety for a special class of differential algebraic cycles is proved. Then as an application, the rigorous definition of differential resultant is given and properties similar to those of the Macaulay resultant are proved. Thirdly, the theory of sparse differential resultants is established, and a single exponential algorithm to compute the sparse differential resultant is given. The concept of Laurent differentially essential systems is introduced and the sparse differential resultant is defined. Then its basic properties are proved. In particular, the concept of differential toric varieties is introduced, and order and degree bounds for the sparse differential resultant are given. Based on these bounds, a single exponential algorithm to compute the sparse differential resultant is proposed.