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The notions of equivalence and strict equivalence for order one differential equations are introduced. The more explicit notion of strict equivalence is applied to examples and questions concerning autonomous equations and equations having the Painleve property. The order one equation determines an algebraic curve. If this curve has genus zero or one, then it is difficult to verify strict equivalence. However, for higher genus strict equivalence can be tested by an algorithm sketched in the text. For autonomous equations, testing strict equivalence and the existence of algebraic solutions are shown to be algorithmic., 19 pages

In this paper we investigate the problem of determining rational parametrizations of plane algebraic curves over an algebraic extension of least degree over the field of definition. This problem reduces to the problem of finding simple points with coordinates in the field of definition on algebraic curves of genus 0. Consequently we are also able to decide parametrizability over the reals. We generalize a classical theorem of Hilbert and Hurwitz about birational transformations. An efficient algorithm for computing such optimal parametrizations is presented.

Let K be a characteristic zero field, let ϕ(t) be a birational parametrization ϕ(t) of a K-definable curve C with coefficients in an algebraic extension K(α) over K. We propose an algorithm to solve the optimization problem of computing the affine reparametrization t→at+b such that ϕ(at+b) has coefficients over an extension of K with algebraic degree as small as possible.