Polynomial vanishing on a finite set of spatial lines
= Polynomial vanishing on a finite set of spatial lines
{title2=$d<4\sqrt L$}
A family of $L\ge1$ lines in $\mathbb R^3$ imposes at most $L(d+1)$ linear conditions on degree-at-most-$d$ polynomials, by <polynomial restriction to a line>. The coefficient space has dimension $\binom{d+3}{3}$. Whenever $(d+2)(d+3)>6L$, a nonzero common vanishing polynomial exists. Taking $d=\lceil\sqrt{6L}\rceil$ gives degree less than $4\sqrt L$.