# Image under “square-morphism”

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?

• Here is a method that works for hypersurfaces. I think one can make a version of it for other varieties but it is computationally intractable. You use Galois theory. Take $2^{n+1}$ copies of $f$, in each of which you multiply a different subset of $(x_0,...,x_n)$ by $-1$. Multiply these all together. This function will have only even powers of $x_i$ in it, so divide all those powers by two. Take the radical if necessary. – Will Sawin Jun 7 '12 at 19:37
• Okay, thank you. Considering symmetry of the $f_1, \ldots , f_r$ could lead to an acceptable result. – Döni Jun 10 '12 at 15:12