Joint of a line collection ($x\in {\ell }_1\cap \cdots \cap {\ell }_n,\mathrm{span}(v_1,\dots ,v_n)={\mathbb{R}}^n$)

For distinct real affine lines in ${\mathbb{R}}^n$, $n\ge 2$, a joint of a line collection is a point lying on $n$ lines with linearly independent direction vectors. In three dimensions the directions must not all lie in one plane. Extra incident lines are allowed; existence of one spanning choice is enough. This counts joint of a line collection points, rather than tuples of lines or incidence multiplicities.

The number of joints of a line collection formed by $L$ distinct real affine lines in ${\mathbb{R}}^n$ is at most $C_n{L}^{n/(n-1)}$. The pruning and minimal-degree polynomial argument proves this bound. The exponent is sharp: the coordinate-line grid has $m^n$ joints of a line collection and $n{m}^{n-1}$ lines. In three dimensions the bound is $C{L}^{3/2}$.

Let a line collection have $M$ joints of a line collection and put $D=\lceil n{M}^{1/n}\rceil$. 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.