  We make a theoretical study of the behavior of a simple fluid displacing a
shear thinning fluid confined in a Hele-Shaw cell. To study the Saffman-Taylor
instability when the displaced fluid is non Newtonian we face the problem of
having a field which is non laplacian. By means of an hodographic
transformation we are able to solve the problem in the case of weak shear
thinning while taking into account the non laplacian character of the equation.
Our results predict that the finger width decreases towards zero for small
values of the surface tension parameter which is inversely proportional to the
finger velocity.
