= Pruning and minimal-degree polynomial argument
Let a line collection have $M$ <joints of a line collection> and put $D=\lceil nM^{1/n}\rceil$. If $M>LD$, deleting each line with at most $D$ current <joints of a line collection>, together with those <joints of a line collection>, leaves a nonempty configuration in which every line has more than $D$ <joints of a line collection>. A nonzero <polynomial> of degree at most $D$ vanishes on the retained <joints of a line collection> by <rank-nullity theorem>. Choose one of minimum degree. Its restriction vanishes identically on every retained line, so its <gradient> is orthogonal to a spanning set of directions at every <joint of a line collection> and hence vanishes there. Each nonzero <partial derivative> would be a lower-degree <polynomial> with the same zeros, contradicting minimality. In characteristic zero, all derivatives vanishing forces a constant <polynomial>, another contradiction. Therefore $M\le LD$, giving the <joints theorem>. The argument's characteristic-zero assumption is essential to its derivative step.
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