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This implies that a system is algebraically integrable (i.e., its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues.
      
Then the ring of invariants ofG is a polynomial ring if and only ifG is generated by pseudoreflections and the pointwise stabilizer inG of any nontrivial subspace has a polynomial ring of invariants.
      
TheS-arithmetic group г is of typeFn, resp.FPn, if and only if for allp inS thep-adic completionGp of the corresponding algebraic groupG is of typeCn resp.CPn.
      
We show that they are induced by automorphisms ofG and that a surjective holomorphic self-map can be nonbijective only in the directions of the nilradical ofG.
      
We classify all instances when a parabolic subgroupP ofG acts on its unipotent radicalPu, or onpu, the Lie algebra ofPu, with only a finite number of orbits.
      
We show that on each Schubert cell, the corresponding Kostant harmonic form can be described using only data coming from the Bruhat Poisson structure.
      
We show that the algebras of semi-invariants of a finite connected quiverQ are complete intersections if and only ifQ is of Dynkin or Euclidean type.
      
We show that in the modular case, the ring of invariants in is of this form if and only if is a polynomial algebra and all pseudoreflections in ?(G) are diagonalizable.
      
We prove that an orbitGυ, υ ∈ V, is polynomially convex if and only ifG?υ is closed andGυ is the real form ofG?υ.
      
Further, an analog of the Kempf-Ness criterion is obtained and homogeneous spaces of compact Lie groups which admit only polynomially convex equivariant embeddings are characterized.
      
Previously, only algorithms for linearly reductive groups and for finite groups have been known.
      
This estimate for multiplicities is uniform, i.e., it depends not on G/H, but only on its complexity.
      
For char k = 0, it is shown that if n is odd, adj(X) is not the product of two noninvertible n × n matrices over k[xij], while for n even, only one special sort of factorization occurs.
      
We show that the structure of a block outside the critical hyperplanes of category O over a symmetrizable Kac-Moody algebra depends only on the corresponding integral Weyl group and its action on the parameters of the Verma modules.
      
An affine pseudo-plane X is a smooth affine surface defined over ${\Bbb C}$ which is endowed with an ${\Bbb A}^1$-fibration such that every fiber is irreducible and only one fiber is a multiple fiber.
      
We consider contractible affine surfaces of negative Kodaira dimension with only quotient singularities.
      
It follows that if such a surface has only one singular point, then it is isomorphic to a quotient C2/G, where G is a finite group acting linearly on C2.
      
Specifically, a subset is complete if and only if it contains infinitely many even-order autocorrelation functions.
      
The constants obtained are independent of the dimension n and depend only on k,p, and the number of different eigenvalues of the matrix B.
      
We require only that the symbols be homogeneous and C2.
      
 

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