When studying local extreme points of functions of several (real) variables, a typical textbook exercise is to consider the polynomial
Here we have and , so the only critical point of is Since and the Hessian of is , it follows that is a local minimum of and, since it is the only critical point, it is in fact an absolute minimum with
being a polynomial, it is reasonable to expect that there is an algebraic explanation as for why is its minimum, and why it lies at . After all, this is what happens in one variable: If and , then
and obviously has a minimum at , and this minimum is
The polynomial of the example above can be analyzed this way as well. A bit of algebra shows that we can write
and it follows immediately that has a minimum value of , achieved precisely when both and , i.e, at
(One can go further, and explain how to go in a systematic way about the `bit of algebra’ that led to the representation of as above, but let’s leave that for now.)
What we did with is not a mere coincidence. Hilbert’s 17th of the 23 problems of his famous address to the Second International Congress of Mathematicians in Paris, 1900, asks whether every polynomial with real coefficients which is non-negative for all (real) choices of is actually a sum of squares of rational functions. (A rational function is a quotient of polynomials.) A nonnegative polynomial is usually called positive definite, but I won’t use this notation here.
If Hilbert’s problem had an affirmative solution, this would provide a clear explanation as for why is non-negative.