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We define the universal exponential extension of an algebraically closed differential field and investigate its properties in the presence of a nice valuation and in connection with linear differential equations. Next we prove normalization theorems for algebraic differential equations over H-fields, as a tool in solving such equations in suitable extensions. The results in this monograph are essential in our work on Hardy fields in 6.
Aschenbrenner et al. (Thu,) studied this question.