ON THE TRANSLATION INVARIANTS OF QUADRATIC DIFFERENTIAL SYSTEMS

	L. A. Timochouk, Delft University of Technology, The Netherlands

	The study of translation invariants of planar autonomous differential
systems with quadratic polynomial right-hand sides seems to be important in
order to make a progress in the general affine-invariant classification of
topological properties of this systems. So far, we obtained the following
results in the specified direction.

   --  For this (graded) algebra of invariants, the annihilating operators
   	are constructed from the Lie equation.

   --	By means of specifically developed software, the linear bases of
   	homogeneous components of the invariant algebra constructed up to
   	degree 5. Besides from 6 trivial invariants of degree 1, there exist
   	8 non-trivial irreducible invariants of degree 3, 3 - of degree 4,
   	ana none of degree 5.

   --	The geometrical meaning of 3rd-degree invariants is revealed.

   --	The absence of so-called "relative invariants" is proved for the
   	shift transformations.

   --  This transformation group acting on the 12-dim coefficient space is
   	proved to be non-reductive, so that the property of its invariant
   	algebra to be finitely generated is not guaranteed by the classical
   	Nagata theorem, and requires a particular proof. That is done using
   	some methods developed by Seshadri, Grosshans, et al.

   --	By construction of the canonical representation of the set of orbits
   	over some generic set of points in R^12, the general algorithm aimed
   	to obtain all the generators of the mentioned algebra has been
   	constructed and run on the basis of "Macaulay" computer algebra system.
   	The new 4 irreducible invariants of degree 6 has been revealed, but
   	the further progress was suspended due to serious computations
   	difficulties.

   --	At the present time, a new software system is being developed (in Ada
   	language), which will hopefully allow to finish construction of the
   	generators of the translation invariant algebra.

The future plans include determination of homological dimension of this
algebra, and the attempt to construct the polynomial basis of the general
affine invariants.