The Main Conjecture for Vinogradov mean values

For integers $s,k\geq 1$, let $J_{s,k}(X)$ count the integral solutions with $1\leq x_i\leq X$ to

$$x_1^j+\cdots+x_s^j=x_{s+1}^j+\cdots+x_{2s}^j,\qquad 1\leq j\leq k.$$

The Main Conjecture. For all integers $s,k\geq 1$,

$$J_{s,k}(X)\ll_{s,k,\epsilon}X^\epsilon\left(X^s+X^{2s-\frac12k(k+1)}\right),$$

for all $X\geq 1$ and every $\epsilon>0$. The two terms correspond respectively to diagonal solutions and the contribution from solutions concentrated near the origin in frequency space. The conjecture was resolved in full in 2015 by Bourgain, Demeter, and Guth, following Wooley's major advances and earlier resolution of the first nontrivial case.