Intersection theory in differential algebraic geometry: Generic intersections and the differential Chow form
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- by Xiao-Shan Gao, Wei Li and Chun-Ming Yuan PDF
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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
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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
- 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).
- © Copyright 2013 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 365 (2013), 4575-4632
- MSC (2010): Primary 12H05, 14C05; Secondary 14C17, 14Q99
- DOI: https://doi.org/10.1090/S0002-9947-2013-05633-4
- MathSciNet review: 3066766