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Displaying similar documents to “Homogeneous approximation in completions of A-fields of non-zero characteristic”

Liouvillian first integrals of homogeneouspolynomial 3-dimensional vector fields

Jean Moulin Ollagnier (1996)

Colloquium Mathematicae

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Given a 3-dimensional vector field V with coordinates V x , V y and V z that are homogeneous polynomials in the ring k[x,y,z], we give a necessary and sufficient condition for the existence of a Liouvillian first integral of V which is homogeneous of degree 0. This condition is the existence of some 1-forms with coordinates in the ring k[x,y,z] enjoying precise properties; in particular, they have to be integrable in the sense of Pfaff and orthogonal to the vector field V. Thus, our theorem...

Extensions of Büchi's problem: Questions of decidability for addition and kth powers

Thanases Pheidas, Xavier Vidaux (2005)

Fundamenta Mathematicae

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We generalize a question of Büchi: Let R be an integral domain, C a subring and k ≥ 2 an integer. Is there an algorithm to decide the solvability in R of any given system of polynomial equations, each of which is linear in the kth powers of the unknowns, with coefficients in C? We state a number-theoretical problem, depending on k, a positive answer to which would imply a negative answer to the question for R = C = ℤ. We reduce a negative answer for k...