Let $f, g_1, \cdots, g_s \in \mathbb{R}[x_1,\cdots,x_n]$ and consider the division of $f$ by the $g_i$. Standard multivariate division algorithm will give $f = \sum_i a_i g_i + r$. I have been trying to understand how the degree of $r$ behaves with respect to the degree of $f$. If $f = \sum_j c_j x^j$, where $j=(j_1,\cdots,j_n)$ is a multiexponent, then the degree of $f$ is defined to be the largest $j_1+\cdots+j_n$.
It seems to me that the degree of $r$ can be as large as the degree of $f$, and that we can't really say anything about it (contrary to the univariate case). Is that right? Is there a way to choose some lexicographic order so that the $\deg(r) < \deg(f)?$
The multivariate division algorithm is not related at all with the degree of the polynomials, rather with a monomial order. If the monomial order is lex, then $X^2Y^5<X^3Y$ but $7>4$. All I can say is that the multivariate division algorithm reduces to the well-known univariate algorithm if consider the monomial order lex. (I guess every monomial order reduces to degree in one variable.)