Hyperbolic Polynomials

Most people who learn about polynomials at some point learn about their roots. So we learn that if you have a polynomial in 1 variable, like $x^2 + 3x + 2$, then the number of complex roots of this polynomial is the same as its degree, but it’s surprisingly a lot harder to say anything about the real roots of a polynomial.

For an arbitrary polynomial, there is a number associated with it called the discriminant. It is defined as [ \prod_{i < j} (r_i - r_j) ] where $r_i$ are the roots of the polynomial. Because this is a symmetric polynomial in the roots, this quantity can also be written as a polynomial in the coefficients of the original polynomial.

For polynomials in degree 2, it looks like [ \Delta(ax^2+bx+c) = b^2-4ac ] For polynomials in degree 3, it looks like [ \Delta(ax^2+bx+c) = b^2-4ac ]

And the discriminant gets a lot more complicated as the degree gets larger.