Division with Remainder in $k[x_1,\ldots,x_2]$

Question

Can you give me a way to established a norm on $k[x_1,\ldots,x_2]$ for division with remainder such in $k[x].$

Thanks.

Answer 1

The more knowledgable commenters apparently won't write an answer, so I will type up a CW summary lest this question end up in the unanswered files.

The answer is 'No'. If a function like a Euclidean norm should exist on an integral domain, then all the ideals of the said ring would have to be principal. But e.g. the ideal $I$ of polynomials $p$ from $k[x,y]$, $k$ any field, such that $p(0,0)=0$ needs two generators, for example $x$ and $y$.

OTOH you can view $k[x_1,x_2]$ as a subring of $k(x_1)[x_2]$. This latter object is the ring of polynomials of $x_2$ with coefficients from the field of rational function $k(x_1)$. Because this is a ring of univariate polynomials with coefficients from a field, it is a Euclidean domain, and hence has a Euclidean norm and a division algorithm. When you apply the division algorithm of this ring to two elements $a,b\neq0$ of the subring $k[x_1,x_2]$ what will often happen is that the division process introduces non-constant polynomials of $k[x_1]$ into the denominators. So the end result (= the $(q,r)$ pair of a quotient and a remainder satisfying the familiar equation $a=qb+r$ with $\deg r < \deg b$) no longer belongs to the subring $k[x_1,x_2]$. Also observe that in the context of the ring $k(x_1)[x_2]$ the degree of a polynomial is determined by looking at powers of $x_2$ only. In this context any power of $x_1$ is just another 'constant coefficient'.

Answer 2

Because $\rm\:k[x_1,\ldots,x_n]\:$ is no longer a PID for $\rm\:n > 1\:,\:$ one requires a generalization of the division algorithm. This is the foundation of various standard basis algorithms, e.g. Gröbner bases, which generalize both the Euclidean algorithm and row reduction. For a very nice introduction see David Bayer's Harvard thesis The Division Algorithm and The Hilbert Scheme, 1982, which was advised by Eisenbud and Mumford. Appended below is an excerpt from the introduction.

Such generalized division algorithms can also be viewed as special cases of critical pair algorithms employed in equational completion "algorithms", e.g. that of Knuth-Bendix. See here for more.