Differential Chow Form for Projective Differential Variety

In this paper, a generic intersection theorem in projective differential algebraic geometry is presented. Precisely, the intersection of an irreducible projective differential variety of dimension d>0 and order h with a generic projective differential hyperplane is shown to be an irreducible projective differential variety of dimension d-1 and order h. Based on the generic intersection theorem, the Chow form for an irreducible projective differential variety is defined and most of the properties of the differential Chow form in affine differential case are established for its projective differential counterpart. Finally, we apply the differential Chow form to a result of linear dependence over projective varieties given by Kolchin.

[1]  W. V. Hodge,et al.  Methods of algebraic geometry , 1947 .

[2]  Bernd Sturmfels,et al.  Product formulas for resultants and Chow forms , 1993 .

[3]  David Eisenbud,et al.  Resultants and Chow forms via exterior syzygies , 2001, math/0111040.

[4]  E. R. Kolchin,et al.  The notion of dimension in the theory of algebraic differential equations , 1964 .

[5]  Ju V Nesterenko.  Estimates for the Orders of Zeros of Functions of a Certain Class and Applications in the Theory of Transcendental Numbers , 1977 .

[6]  Teresa Krick,et al.  The Computational Complexity of the Chow Form , 2002, Found. Comput. Math..

[7]  W. Brownawell.  Bounds for the degrees in the Nullstellensatz , 1987 .

[8]  E. Kolchin.  Differential Algebra and Algebraic Groups , 2012 .

[9]  Xiao-Shan Gao,et al.  Sparse differential resultant , 2011, ISSAC '11.

[10]  I. M. Gelʹfand,et al.  Discriminants, Resultants, and Multidimensional Determinants , 1994 .

[11]  W. L. Edge.  Methods of Algebraic Geometry. By W. V. D. Hodge and D. Pedoe. Vol. I: pp. viii, 440; Vol. II: pp. ix, 394. Paperback edition. 17s. 6d. each volume. (Cambridge.) , 1970 .

[12]  Hyman Bass,et al.  Selected works of Ellis Kolchin with commentary , 1999 .

[13]  Hlawka.  Einführung in die algebraische Geometrie , 1941 .

[14]  Patrice Philippon,et al.  Critères Pour L’indépendance Algébrique , 1986 .

[15]  Xiao-Shan Gao,et al.  Intersection theory in differential algebraic geometry: Generic intersections and the differential Chow form , 2010, Transactions of the American Mathematical Society.