Sytem of Dynamic and Differential Physics Kentu Notes-138.pdf

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266 L. Stolovitch that O S n = C [[ u 1 ,..., u p ]] . Let π : C n C p defined by π ( x ) = ( u 1 ( x ) ,..., u p ( x )) . Let s be the degree of transcendence of the field of fractions of C [ u 1 ,..., u p ] ; it is the maximal number of algebraically independent polynomials among u 1 ,..., u p . The algebraic relations among u 1 ,..., u p define an s -dimensional algebraic variety C S in C p . Hence, π defines a singular fibration over C S . The linear vector fields S 1 ,..., S l are tangent and independent on each fiber π 1 ( b ) of π ; the latter are called toric va- riety because they admit an action of the algebraic torus C . Note that we must have l n s . Now, we come to the nonlinear deformation. Let X = S + ε be a nonlinear deformation of S . Let us assume that it is formally completely integrable. Then, ac- cording to our result, there exists a neighborhood U of 0 in C n and an holomorphic diffeomorphism Φ on U such that, in the new coordinate system, the vector fields Φ X 1 ,..., Φ X l are commuting linear diagonal vector fields on each fiber restricted to U and their eigenvalues depend only on the fiber. Indeed, in this new coordinates, we have Φ X i = l j = 1 a i , j S j where a i , j O S n . By definition, these vector fields are all tangent to the fibers of π (therefore, we must have l n s ). As consequence Φ X i ’s are all tangent to the fibers of π . On each fiber, the functions a i , j are constant so that each Φ X i
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