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Simple Differentially Definable Functions

D-finite functions satisfy linear differential equations with polynomial coefficients. The solutions to this type of equations may have singularities determined by the zeros of their leading coefficient. There are algorithms to desingularize the equations, i.e., remove singularities from the equation that do not appear in its solutions. However, classical computations of closure properties (such as addition, multiplication, etc.) with D-finite functions return equations with extra zeros in the leading coefficient. In this paper we present theory and algorithms based on linear algebra to control the leading coefficients when computing these closure properties and we also extend this theory to the more general class of differentially definable functions.