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

@article{Gao2010IntersectionTI, title={Intersection theory in differential algebraic geometry: Generic intersections and the differential Chow form}, author={X. Gao and Wei Li and Chunming Yuan}, journal={arXiv: Algebraic Geometry}, year={2010} }

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…

## 32 Citations

Differential Chow Form for Projective Differential Variety

- MathematicsArXiv
- 2011

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 projectives differential counterpart.

Bertini theorems for differential algebraic geometry

- Mathematics
- 2012

We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a…

Differential Chow varieties exist

- MathematicsJ. Lond. Math. Soc.
- 2017

In the course of the proof several definability results from the theory of algebraically closed fields are required, and elementary proofs of these results are given.

Computation of differential Chow forms for ordinary prime differential ideals

- Mathematics, Computer ScienceAdv. Appl. Math.
- 2016

Differential elimination by differential specialization of Sylvester style matrices

- MathematicsAdv. Appl. Math.
- 2016

Partial differential Chow forms and a type of partial differential Chow varieties

- MathematicsCommunications in Algebra
- 2020

Abstract We first present an intersection theory of irreducible partial differential varieties with quasi-generic differential hypersurfaces. Then, we define partial differential Chow forms for…

Matrix Formulae of Differential Resultant for First Order Generic Ordinary Differential Polynomials

- MathematicsASCM
- 2012

The algebraic sparse resultant of the differential resultant of \(f_1, f_2, {\updelta } f_1\) and \({\ updelta ) f-2\) treated as polynomials in \(y, y, y^{\prime }, y^{prime \prime }\) is shown to be a nonzero multiple of the equations.

PARTIAL DIFFERENTIAL CHOW FORMS AND A TYPE OF PARTIAL DIFFERENTIAL CHOW VARIETIES

- Mathematics
- 2017

We first present an intersection theory of partial differential varieties with quasi-generic differential hypersurfaces. Then, based on the generic differential intersection theory, we define the…

## References

SHOWING 1-10 OF 60 REFERENCES

The notion of dimension in the theory of algebraic differential equations

- Mathematics
- 1964

with coefficients in a differential field ^ (ordinary or partial) ; here 2 is any subset of the differential polynomial algebra & = ${yi, • • • , yn} over 5\ Denote the set of all solutions of this…

Specializations in differential algebra

- Mathematics
- 1959

1. Objectives and summary. Much of elementary differential algebra can be regarded as a generalization of the algebraic geometry of polynomial rings over a field to an analogous theory for rings of…

Resolvent Representation for Regular Differential Ideals

- MathematicsApplicable Algebra in Engineering, Communication and Computing
- 2003

Abstract We show that the generic zeros of a differential ideal [A]:H∞A defined by a differential chain A are birationally equivalent to the general zeros of a single regular differential polynomial.…

Selected works of Ellis Kolchin with commentary

- Mathematics
- 1999

Picard-Vessiot theory of partial differential fields The notion of dimension in the theory of algebraic differential equations Part I. The Papers of Ellis Kolchin: On certain ideals of differential…

The Computational Complexity of the Chow Form

- Mathematics, Computer ScienceFound. Comput. Math.
- 2004

A bounded probability algorithm for the computation of the Chowforms of the equidimensional components of an algebraic variety that improves (or meets in some special cases) the complexity of all previous algorithms for computing Chow forms.

Differential Equations in a Projective Space and Linear Dependence over a Projective Variety

- Mathematics
- 1974

Resultants and Chow forms via exterior syzygies

- Mathematics
- 2001

Let W be a vector space of dimension n+ 1 over a field K. The Chow divisor of a k-dimensional variety X in P = P(W ) is the hypersurface, in the Grassmannian Gk+1 of planes of codimension k+1 in P,…

Heights of projective varieties and positive Green forms

- Mathematics
- 1994

Using arithmetic intersection theory, a theory of heights for projective varieties over rings of algebraic integers is developed. These heights are generalizations of those considered by Weil,…

Resultants and the algebraicity of the join pairing on Chow varieties

- Mathematics
- 1997

The Chow/Van der Waerden approach to algebraic cycles via resultants is used to give a purely algebraic proof for the algebraicity of the complex suspension. The algebraicity of the join pairing on…

Order and dimension

- Mathematics
- 1983

It will be shown that the conjectured Jacobi bound for the order of differential systems cannot be valid if a very natural conjecture concerning the differential dimension of such systems is false.…