Suppose $X$ is a finite set of points in $\mathbb C^n$. Let $d\_r$ denote the minimum degree of a polynomial vanishing to order $r$ at each point of $X$. By linear algebra, we know find can find a polynomial of degree $d$ vanishing to order $r$ on $X$ provided $\binom{r+n-1}{n}|X|<\binom{n+d}{d}$. Now, suppose $d\_r$ is much smaller than this for some $r>1$. Does this fact tell us $d\_1$ must be small?



Degrees of polynomials vanishing to various orders on a set of points
There is a conjecture of Chudnovsky that 
$$\frac{d\_r}{r}\geq \frac{d\_1+n-1}{n}$$
and I believe the best current bound is the one by Esnault and Viehweg



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> "Sur une minoration du degre d’hypersurfaces s’annulant en certains points" Math. Ann. 263 (1983), no. 1, 75–86
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where it is proved that $$\frac{d\_r}{r}\geq \frac{d\_1+1}{n}.$$
Either way, the answer is yes: $d\_1$ is bounded above by a simple function of $d\_r$. For a summary of work in this direction look at the ["Regina Lectures on Fat Points"](https://arxiv.org/abs/1310.3552) by Cooper and Harbourne.


