Proposition

Let $I = \ev{ x^6, x^2y^3, xy^7 } \subset k[x, y]$.

  1. In the $(m, n)$-lane, plot the set of exponent vectors $(m, n)$ of monomials $x^my^n$ appearing in elements of $I$.
  2. If we apply the division algorithm to an element $f \in k[x, y]$, using the generators of $I$ as divisors, what terms can appear in the remainder?

Solution

Plot Therefore, any monomial $x^my^n$ such that

  • $m = 0$, or
  • $m = 1$ and $n \leq 6$, or
  • $2 \leq m \leq 5$ and $n \leq 2$.

can appear in the remainder.