Embed some combinatorial problem into a vector space.
Capture the hypotheses of the problem by constructing a polynomial of low-degree that is zero on a certain set
After constructing the polynomial, argue about its algebraic properties to deduce that the original configuration must satisfy the desired properties.
The [3] by Dvir
The problem by Ellenberg and Gijswijt[7] with the original framework developed on the analogous problem over Z 4 n {\displaystyle \mathbb {Z} _{4}^{n}} by Croot, Lev and Pach[8]
The by Guth and Katz[4]
The Joints Problem in 3D by Guth and Katz. Their argument was later simplified by Elekes, Kaplan and Sharir[10]
A few examples of longstanding open problems that have been solved using the polynomial method are:
by Terence Tao
by Larry Guth
