Hi all I was given this question and desperately in need of help. I am given a homogeneous polynomial of degree 4 of two variables x and y, with real coefficients with 4 real distinct projective roots i.e. functions $ y = \alpha_ix $ such that substituting in the polynomial gives identically zero function. I am asked to show via change of coordinates of x and y to new coordinates (of course non singular with non singular Jacobian) that in the new coordinates I have the polynomial turns into one of $ x^4 + ax^2y^2+y^4 ; a<-2 $ OR $ -x^4 + ax^2y^2-y^4 ; a>2 $ I have no real idea how to do this as to what change to find and I am in desperate need for some help all appreciated thanks
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