What is the general definition of a discriminant? (Not just the definition for polynomials) For example, in regards to the second derivative test for a function of two variables, $D=f_{xx}f_{yy}-(f_{xy})^2$ is refered to as the "second derivative test discriminant."
I know that D is the determinant of $\pmatrix{f_{xx}&f_{xy}\\f_{yx}&f_{yy}}$, but I do not know the significance of this or how it relates to the polynomial definition of a discriminant.
 A: Discriminants have their roots in abstract algebra and are used to determine the nature of zeroes of polynomials. From Wolfram Mathworld, if $p(z)$ is an $n$th degree polynomial with leading coefficient $a_n$ and roots $r_i$, the discriminant $\Delta$ is
$$\Delta(p) = a_n^{2n-2} \prod_{i > j} (r_i - r_j)^2.$$
Discriminants are defined up to a constant factor, so the definition could just be taken as the product of the squares of the differences. As written, the formula is useless because discriminants are meant to provide information about the roots given the coefficients of the polynomial. The horrendous formulas for higher degree polynomials in terms of coefficients  is actually a beautiful result from some clever field theory.
Discriminants only refer to and should only refer to the aforementioned property of polynomials and similar structures/generalizations (conic sections is a classic example). The name 'second derivative test discriminant' is rather unfortunate -- chances are, it takes its name from either the similar-looking quadratic discriminant or the purpose it serves in the second derivative test (in the sense that its sign helps determine the nature of the critical point). Another reason the name is misleading is that the sign of the 'second derivative test discriminant' does not give complete information about the critical point (for quadratics, it does). The purpose of computing the determinant is to check if the matrix is positive or negative definite, and if it is positive, we do not know if the point is a local minimum or maximum unless we know the sign of $f_{xx}$.
A: The name comes from the notion of discriminant of a quadratic form of two variables and Taylor's formula at order $2$:
\begin{multline}f(x_0+h,y_0+k=f(x_0,y_0)+f'_x(x_0,y_0)h+f'_y(x_0,y_0)k\\+\frac12\bigl(f''_{xx}(x_0,y_0)h^2+2f''_{xy}(x_0,y_0)hk+f''_{yy}(x_0,y_0)k^2\bigr)+o\bigl(\lVert(h,k)\rVert^2\bigr)\end{multline}
This discriminant is the reduced discriminant of the quadratic polynomial $f''_{xx}T^2+2f''_{xy}T+f''_{yy}$, and its sign is a criterion to determine if the quadratic form is (definite) positive, (definite) negative or none, which in turn is interpreted as $f$ having a local minimu, a local minimum or a saddle point.
