  Consider a polynomial $F$ in $m$ variables and a finite point ensemble $S=S_1
\times ... \times S_m$. When given the leading monomial of $F$ with respect to
a lexicographic ordering we derive improved information on the possible number
of zeros of $F$ of multiplicity at least $r$ from $S$. We then use this
information to design a list decoding algorithm for a large class of affine
variety codes.
