# Pure to Applied Math: Self-driving Cars & “Sum of 2 Squares” Polynomial

Key Points:

• 1900 Hilbert’s 17th Conjecture: Non-negative Polynomial <=> sum of 2 squares (Proved by Emile Artin in 1927)
• Computing Math : approximate by optimisation with “Linear Programs” which are faster to compute.
• Princeton Mathematicians applied it to self-driving cars.

https://www.wired.com/story/a-classical-math-problem-gets-pulled-into-self-driving-cars/amp

Explain:

Sum of 2 Squares <=> always non-negative ( 0)

$13 = 4 + 9 = 2^{2} + 3^{2}$

$P (x) = 5x^2+16x+13 = (x+2)^{2} + (2x+3)^{2} \geq 0$

Self-driving Car: Trajectory = P (x)

P(x) < 0 where the car’s position in the trajectory;

Obstacles are positions where P (x) 0.

This is one of the many cases of Pure Math turned to be Applied Math in last few decades. Other examples:

Is Applied Math => Pure Math ?
Yes!