Recovery of Algebraic Numbers from their $p$-adic Approximations

We describe three ways to generalise Lenstra's algebraic integer recovery method. One direction adapts the algorithm so that rational numbers are automatically produced given only upper bounds on the sizes of the numerators and denominators. Another direction produces a variant which recovers algebraic numbers as elements of multiple generator algebraic number fields. The third direction explains how the method can work if a reducible minimal polynomial had been given for an algebraic generator. Any two or all three of the generalisations may be employed simultaneously.