Every cubic form over integers in 14 variables has a non-trivial zero

You need to know: Polynomial in n variables.

Background: A cubic form over integers in n variables is the polynomial of the form $P(x_1, x_2, \dots, x_n)=\sum\limits_{i=1}^n \sum\limits_{j=1}^n \sum\limits_{k=1}^n c_{ijk}x_ix_jx_k$ , where $c_{ijk}$ are integer coefficients.

The Theorem: On 8th June 2006, Roger Heath-Brown submitted to Inventiones mathematicae a paper in which he proved that, for every cubic form $P(x_1, x_2, \dots, x_n)$ over integers in $n\geq 14$ variables, there exists integers $x^*_1, x^*_2, \dots, x^*_n$ , not all zero, such that $P(x^*_1, x^*_2, \dots, x^*_n)=0$ .

Short context: The classical 1884 Theorem of Meyer states that any indefinite quadratic from (a quadratic from $P(x_1, x_2, \dots, x_n)=\sum\limits_{i=1}^n \sum\limits_{j=1}^n c_{ij}x_ix_j$ is indefinite if it is less than $0$ for some values of variables and greater than $0$ for others) over integers in $n\geq 5$ variables has a non-trivial zero. All cubic forms are indefinite, and it is conjectured that they must have a non-trivial zero if $n \geq 10$ . In 1963, Davenport proved this for $n\geq 16$ . After more then 40 years with no further progress, the Theorem proves this for $n\geq 14$ .