Suppose that we have a system of polynomial equations that looks like, say,
$P_1(x) = a_1 x_1 + \cdots + a_n x_n \\ P_2(x) = b_1 x_1^2 + \cdots + b_n x_n^2 + b_{n+1} x_1 + \cdots + b_{2n} x_n\\ P_3(x) = c_1 x_1^3 + \cdots + c_{3n} x_n\\ \vdots \\ P_n$
and so on. We can also consider the system of equations formed by the terms of highest total degree:
$\tilde{P}_1(x) = P_1(x)\\ \tilde{P}_2(x) = b_1 x_1^2 + \cdots + b_n x_n^2 \\ \tilde{P}_3(x) = c_1 x_1^3 + \cdots + c_n x_n^3 \\ \vdots \\ \tilde{P}_n$
By the Finiteness Theorem of algebraic geometry we know that if the latter system has a finite number of solutions then for each $i$, $1\le i \le n$, there is an $m_i \ge 0$ such that $x_i^{m_i} = \mathrm{LT}(g)$ for some $g = Q_1 \tilde{P}_1 + \cdots + Q_n \tilde{P}_n$. Assuming that we use graded lexicographic order (meaning that terms with highest total degree have the highest order overall) this seems to imply that $x_i^{m_i} = \mathrm{LT}( Q_1 P_1 + \cdots + Q_n P_n)$ so that $x_i^{m_i} \in \mathrm{LT} \langle \{ P_i \} \rangle$ as well. This would ostensibly imply that $\{ P_i \}$ has a finite number of solutions as well. So can we use the reduced system of leading terms to bound the number of solutions to the full unreduced system in this way?