Differential Chow form and sparse differential resultant

Abstract

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.