Suppose $R$ is a subalgebra of ${\mathbb C}[x_1,...,x_N]$ generated by polynomials $p_1,...,p_k.$ I know that ${\mathbb C}[x_1,...,x_N]$ is the integral closure of $R$. Is there an algorithm to determine if $R={\mathbb C}[x_1,...,x_N]$ which would work for large $N$? (Say $N=20$).

Clearly, it is a question about the normality of the variety $X=Spec(R).$ The problem is that I don't know relations between the generators of $R$ (and these may be difficult to compute for large $N$)-- otherwise I could check if $X$ is smooth and if the map ${\mathbb C}^n\to X$ is an immersion -- conditions which imply that $R={\mathbb C}[x_1,...,x_N]$.