Abstract Algebra – Division Algorithm for Multivariate Polynomials

Question

We know that if $F$ is a field and $f(X)$ a non-zero polynomial in $F[X]$, then for every polynomial $g(X)$ we can find $q(X),r(X)$ such that
$$g(X)=f(X)\cdot q(X)+r(X)$$
with $r(X)$ the zero polynomial or $\deg r(X)<\deg f(X)$.
My question is: the same algorithm holds for many variables? So can i write
$$g(X_1,\ldots,X_n)=f(X_1,\ldots,X_n)\cdot q(X_1,\ldots,X_n)+r(X_1,\ldots,X_n)$$

Answer 1

No, the polynomial division algorithm does not immediately generalize to multivariate rings. Here is a simple proof. Just as for $\rm\:\Bbb Z,\:$ a domain having an algorithm for division with smaller remainder, also enjoys Euclid's algorithm for gcds, which, in extended form, yields Bezout's identity. Therefore gcds have linear representation $\rm\:gcd(a,b) = r a + s b\:$ (i.e. Euclidean domains are Bezout). But this fails in multivariate polynomial rings $\rm\:F[x_1,\ldots, x_n],\ n \ge 2,\:$ since $\rm\:gcd(x_1,x_2) = 1\:$ but there is no Bezout equation $\rm\: 1 = x_1 f + x_2 g\ $ (evaluating at $\rm\:x_1 = 0 = x_2\,$ $\Rightarrow$ $\rm\,1 = 0\:$ in $\rm\,F,\,$ contra field axioms).

But all Euclidean is not lost, since one can generalize the polynomial division algorithm in a way that recovers many of the important properties. For this, look up the Grobner basis algorithm, which may be viewed as a nonlinear generalization of the Euclidean/division algorithm and Gaussian elimination.