Polynomial stability of differential fields

A notion of complexity for an arbitrary structure was defined in the book o f Poizat Les petits cailloux (1995): we can define P and NP problems over a differential field K. Using the Witness Theorem of Blum et al., we prove t he P-stability of the theory of differential fields: a P problem over a dif ferential field K is still P when restricts to a sub-differential field It of K. As a consequence, if P = NP over some differentially closed field K, then P = NP over any differentially closed field and over any algebraically closed field.