We consider systems of ordinary differential equations which are polynomial in the unknown functions and their derivatives. For a given system, we are concerned with computing algebraic constraints on the initial conditions such that on the algebraic variety determined by the constraint equations, the initial value problem of the original system of differential equations has a unique solution. For these systems, we introduce the concept of essential degree, and an algorithmic process called prolongation. The prolongation process may be repeated at most a finite number of times, at the end of which the original system is replaced by an equivalent system (that we called complete). The length of the prolongation process is an invariant which we call the algebraic index of the system. Using basic transformations, we reduce our study to quasi-linear systems, for which we prove an existence and uniqueness theorem, which identifies the algebraic constraints on initial conditions. Both over-determined and under-determined systems are studied. In the case of quasi-linear systems, algorithms are developed and implemented.