# Computing the ideal of relations

Given two projective spaces $\mathbb P^n$ and $\mathbb P^m$ together with $m+1$ global sections of the invertible sheaf $\mathcal O_{\mathbb P^n}(d)$ (e.g. $m+1$ homogeneous polynomials of degree $d$ in the variables $x_0,\cdots,x_n$, say $f_0,\cdots,f_m$), we know that there exists a unique morphism $[f_0,\cdots,f_m] : \mathbb P^n \to \mathbb P^m$. Assume the projective spaces are considered over a noetherian ring ; the morphisms to the base are both projective, hence proper, which means $[f_0,\cdots,f_m]$ is a proper morphism, hence has closed image.

Question : Does there exist an algorithm already implemented in Sage to find the homogeneous ideal of relations of the image of the map $[f_0,\cdots,f_m]$? I've been messing around for a few days now and it seems to only involve linear algebra, so in the case where the base is the spectrum of a field there should be an algorithm, I just don't know how efficient it is or if it's implemented at all. I would not mind if the algorithm was slow, I just want it to work in small cases (i.e. small degree and small number of polynomials)!