Weighted monic relations yield eventual bounded representatives

Weighted monic relations in a power of an ideal produce eventual bounded-support representatives modulo every power of the ideal, with any degree rate above the relations' rate.

Let $I\subseteq\mathbb{C}[X_1,\dots,X_d]$ be an ideal, $w_i>0$ rational degree weights. Suppose that for some $k>0$ and positive integers $b_1,\dots,b_d$ there are polynomials $F_i\in I^k$ such that every exponent $v$ in the support of the monic relation $X_i^{b_i}-F_i$ satisfies

and the relation degrees obey a rate budget

for some rational $a>0$. Then for every rational rate $\delta>a$ there is a threshold $n_0$ such that every polynomial $p$ has, for all $n\ge n_0$, a representative $q$ with

The proof is weighted monic division: a monomial $X^u$ whose weighted degree exceeds $n\delta$ has floor cost $\sum_i \lfloor u_i/b_i\rfloor$ exceeding $n/k$, so a product of $n/k$ relations gives an element of $I^n$ agreeing with $X^u$ on its leading monomial and strictly decreasing the weighted degree; well-foundedness of weighted-degree descent then yields remainders supported below weighted degree $n\delta$, and linearity extends the construction from monomials to all polynomials.