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Rational Functions on Algebraic Sets
Vector Fields Associated to an Irreducible
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19-th century abelian algebraic correspondence algebraic geometry algebraic sets algebraically closed algebro-geometric apply basic classical coefficients common zero complex numbers Consider controllable curve defined Definition derivation deterministic differential equation differential geometry dimension dual eigenvalues element example Exercise F e L(X,U feedback group following conditions formula given GL(X Hamiltonian Hence Hermann image set input input-output system irreducible algebraic set K-1inear Kronecker Lie algebra Lie group linear map linear subspace linear systems manifold matrix non-zero observable one-one orbit output PF(X PF(Y pole placement polynomial functions polynomial map problem Proof properties prove rational functions rational map relation Remark result scalar field semidirect product solution stochastic subgroup submersion Suppose symmetric Systems Theory tangent space tangent vectors techniques Theorem 6.1 tion topology transformation group vanish variables vector field vector space Zariski closed subset Zariski open Zariski open subset