Degree of the divisor of solutions of a differential equation on a projective variety

Using the data schemes developed by Arrondo-Sols-Speiser, we give a rigorous definition of algebraic differential equations on the complex projective space $P^n$. For an algebraic subvariety $S \subseteq P^n$, we present an explicit formula for the degree of the divisor of solutions of a differential equation on $S$ and give some examples of applications. We extend the technique and result to the real case.