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ANALOGIES BETWEEN ALGEBRAIC AND DIFFERENTIAL
Rational Functions on Algebraic Sets
Vector Fields Associated to an Irreducible
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19-th century abelian algebraic geometry algebraic sets algebraic subset algebro-geometric apply basic coefficients common zero complex numbers Consider contains a Zariski controllable curve defined Definition denote differential equation differential geometry dimension dual eigenvalues element elimination theory Exercise F e L(X,U feedback group following conditions g e G group action Hamiltonian Hence homogeneous polynomials image set input irreducible algebraic set Kronecker Lie algebra Lie group linear map linear subspace linear systems Liouville transformation manifold mathematical matrix non-zero observable one-one orbit output parameter PF(X polynomial function polynomial map problem Proof properties prove PS(B rational functions rational map relation Remark RF(Y right hand side scalar field semidirect product solution solvability state-feedback equivalent stochastic submersion symmetric Systems Theory tangent space tangent vector techniques Theorem 6.1 tion transformation group vanishing variables vector field vector space Zariski closed subset Zariski open subset