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

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 the 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.

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

[2]  Giuseppa Carrà Ferro,et al.  A Resultant Theory for the Systems of Two Ordinary Algebraic Differential Equations , 1997, Appl. Algebra Eng. Commun. Comput..

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

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

[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]  Xiao-Shan Gao,et al.  A Zero Structure Theorem for Differential Parametric Systems , 1993, J. Symb. Comput..

[7]  B. L. Waerden,et al.  Einführung in die algebraische Geometrie , 1973 .

[8]  J. Todd.  Methods of Algebraic Geometry , 1948, Nature.

[9]  David A. Cox,et al.  Using Algebraic Geometry , 1998 .

[10]  Resultants and the algebraicity of the join pairing on Chow varieties , 1997 .

[11]  J. Ritt,et al.  Differential Equations from the Algebraic Standpoint , 1933, Nature.

[12]  J. Rafael Sendra,et al.  Linear complete differential resultants and the implicitization of linear DPPEs , 2010, J. Symb. Comput..

[13]  E. R. Kolchin,et al.  Extensions of Differential Fields, II , 1942 .

[14]  Marc Moreno Maza,et al.  Computing differential characteristic sets by change of ordering , 2010, J. Symb. Comput..

[15]  Evelyne Hubert,et al.  Resolvent Representation for Regular Differential Ideals , 2003, Applicable Algebra in Engineering, Communication and Computing.

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

[17]  William Y. Sit.  THE RITT–KOLCHIN THEORY FOR DIFFERENTIAL POLYNOMIALS , 2002 .

[18]  A. Seidenberg.  An elimination theory for differential algebra , 1959 .

[19]  J. Jouanolou,et al.  Le formalisme du résultant , 1991 .

[20]  A. V. Mikhalev,et al.  Differential and Difference Dimension Polynomials , 1998 .

[21]  Oleg Golubitsky,et al.  Algebraic transformation of differential characteristic decompositions from one ranking to another , 2009, J. Symb. Comput..

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

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

[24]  Jean-Benoît Bost,et al.  Heights of projective varieties and positive Green forms , 1994 .

[25]  F. Rouillier.  Solving Zero-dimensional Polynomial Systems through the Rational Univariate Representation , 1998 .

[26]  A. Rosenfeld.  Specializations in differential algebra , 1959 .

[27]  Wenjun Wu,et al.  Basic principles of mechanical theorem proving in elementary geometries , 1986, Journal of Automated Reasoning.

[28]  E. R. Kolchin.  Extensions of differential fields. III , 1947 .

[29]  Giuseppa Carrà Ferro.  A Resultant Theory for Ordinary Algebraic Differential Equations , 1997, AAECC.

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

[31]  Brahim Sadik,et al.  A Bound for the Order of Characteristic Set Elements of an Ordinary Prime Differential Ideal and some Applications , 2000, Applicable Algebra in Engineering, Communication and Computing.

[32]  Xiao-Shan Gao,et al.  Automated reasoning in differential geometry and mechanics using the characteristic set method , 1993, Journal of Automated Reasoning.

[33]  Fabrice Rouillier,et al.  Solving Zero-Dimensional Systems Through the Rational Univariate Representation , 1999, Applicable Algebra in Engineering, Communication and Computing.

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

[35]  A. Seidenberg.  On the Chow form , 1975 .

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

[37]  Leandro Caniglia,et al.  How to compute the Chow form of an unmixed polynomial ideal in single exponential time , 1990, Applicable Algebra in Engineering, Communication and Computing.

[38]  R. Cohn.  Order and dimension , 1983 .

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

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