Let $V \subset \mathbb{P}^n$ be a projective variety. The homogeneous vanishing ideal is generated by the forms $f_1, \ldots , f_r$. Now consider the morphism

$\phi: \mathbb{P}^n \to \mathbb{P}^n, (x_0 : \ldots : x_n) \mapsto (x_0^2: \ldots : x_n^2).$

Is there a way to find generators of the vanishing ideal of the image $\phi(V$) under this morphism, i.e. an easier or more conceptional way than computung the image via Groebner bases?