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Restricting calculations to the last factorization

x,y,α,β=var('x,y,α,β')
P.<U,V> = PolynomialRing(SR, 'U,V')
hes0=-(V^2*y^2*α^2 - 2*U*V*x*y*α*β + U^2*x^2*β^2 - V^2*y^2*α - U^2*x^2*β)
hes0

(-x^2*β^2 + x^2*β)*U^2 + 2*x*y*α*β*U*V + (-y^2*α^2 + y^2*α)*V^2

Q = QuadraticForm.from_polynomial(hes0);
mG=Q.Gram_matrix();
mm=mG.apply_map(factor);mm

[-x^2*(β - 1)*β        x*y*α*β]
[       x*y*α*β -y^2*(α - 1)*α]

mm.det().factor()

-x^2*y^2*(α + β - 1)*α*β

one can observe that α,β in (0,1) doesn't imply definitness