Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Intersection theory in differential algebraic geometry: Generic intersections and the differential Chow form


Authors: Xiao-Shan Gao, Wei Li and Chun-Ming Yuan
Journal: Trans. Amer. Math. Soc. 365 (2013), 4575-4632
MSC (2010): Primary 12H05, 14C05; Secondary 14C17, 14Q99
Published electronically: February 12, 2013
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, an intersection theory for generic differential polynomials is presented. The intersection of an irreducible differential variety of dimension $ d$ and order $ h$ with a generic differential hypersurface of order $ s$ is shown to be an irreducible variety of dimension $ d-1$ and order $ h+s$. As a consequence, the dimension conjecture for generic differential polynomials is proved. Based on intersection theory, 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. Furthermore, the generalized differential Chow form is defined and its properties are proved. As an application of the generalized differential Chow form, the differential resultant of $ n+1$ generic differential polynomials in $ n$ variables is defined and properties similar to that of the Macaulay resultant for multivariate polynomials are proved.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 12H05, 14C05, 14C17, 14Q99

Retrieve articles in all journals with MSC (2010): 12H05, 14C05, 14C17, 14Q99


Additional Information

Xiao-Shan Gao
Affiliation: KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
Email: xgao@mmrc.iss.ac.cn

Wei Li
Affiliation: KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
Email: liwei@mmrc.iss.ac.cn

Chun-Ming Yuan
Affiliation: KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
Email: cmyuan@mmrc.iss.ac.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05633-4
PII: S 0002-9947(2013)05633-4
Keywords: Differential Chow form, differential Chow variety, differential resultant, dimension conjecture, intersection theory, differential algebraic cycle, differential Stickelberger’s Theorem, generic differential polynomial.
Received by editor(s): August 19, 2010
Received by editor(s) in revised form: May 7, 2011
Published electronically: February 12, 2013
Additional Notes: This work was partially supported by a National Key Basic Research Project of China (2011CB302400) and by a grant from NSFC (60821002).
Article copyright: © Copyright 2013 American Mathematical Society