Let $I_X\subseteq \mathbb{C}[x_0,\ldots,x_n]$ be the homogeneous vanishing ideal of a set $X$ of $s$ points in $\mathbb{P}^2$. Let $d_1$ resp. $d_2$ denote the minimal resp. maximal degree of elements in a minimal set of generators of $I_X$. We always have $d_2\leq s$, but unless $X$ is contained in a line (i.e. $d_1=1$) this bound is not sharp. By Bezout's Theorem we have that $d_2\geq \frac{s}{d_1}$, so there is a lower bound on $d_2$ that is getting smaller for growing $d_1$. Is there something similar known for upper bounds, i.e. :

What is the best known upper bound on $d_2$ depending only on $s$ and $d_1$?


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