= Finite flat Noether normalization of an affine hypersurface
{title2=$X\to\mathbb A_k^{n-1}\text{ finite and flat}$}
For a nonconstant polynomial over any <field>, use triangular substitutions $x_i=y_i+y_n^{B^i}$, $x_n=y_n$, where $B$ exceeds every exponent in its support. Base-$B$ weights distinguish all original <monomials>, so the transformed polynomial has a unique highest $y_n$ power with coefficient in $k^\times$. Rescale to make it monic. The <monic polynomial quotient is finite free> over $k[y_1,\ldots,y_{n-1}]$, yielding a finite flat projection. The nonlinear substitution works over <finite fields> as well as infinite ones; a generic linear-direction argument alone would not do so.
Back to article page
