Joints tightened

Given a set of lines in R3, a joint is a point passed through by three lines not all lying in some plane. The joints problem asks for the maximum number of joints formed by L lines. Using the polynomial method, Guth and Katz proved that the answer is O(L3/2), which is best possible up to a root constant factor. We prove a new upper bound of ( 2/3)L3/2, which matches, up to a 1 + o(1) factor, the best known construction: place k generic planes, and use their pairwise intersections to form (k) lines and their triple-wise intersections to form (k) joints. Guth conjectured that this construction is 2 3 optimal. Our result generalizes to arbitrary dimensions and fields. Our technique builds on the work on Ruixiang Zhang proving the multijoints conjecture via an extension of the polynomial method. We set up a variational problem to control the high order of vanishing of a polynomial at each joint.