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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.Comment: 17 page
Computing the Chow variety of quadratic space curves
Quadrics in the Grassmannian of lines in 3-space form a 19-dimensional
projective space. We study the subvariety of coisotropic hypersurfaces.
Following Gel'fand, Kapranov and Zelevinsky, it decomposes into Chow forms of
plane conics, Chow forms of pairs of lines, and Hurwitz forms of quadric
surfaces. We compute the ideals of these loci
Multiplicity estimates, analytic cycles and Newton polytopes
We consider the problem of estimating the multiplicity of a polynomial when
restricted to the smooth analytic trajectory of a (possibly singular)
polynomial vector field at a given point or points, under an assumption known
as the D-property. Nesterenko has developed an elimination theoretic approach
to this problem which has been widely used in transcendental number theory.
We propose an alternative approach to this problem based on more local
analytic considerations. In particular we obtain simpler proofs to many of the
best known estimates, and give more general formulations in terms of Newton
polytopes, analogous to the Bernstein-Kushnirenko theorem. We also improve the
estimate's dependence on the ambient dimension from doubly-exponential to an
essentially optimal single-exponential.Comment: Some editorial modifications to improve readability; No essential
mathematical change
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