What does happen when I am studying and classifying the critical points of a multivariabile function in the case of a zero determinant of the Hessian matrix?
I have searched over here, and on books, but the phrase that always comes out is something like
$$\text{In this case the test is undefined, and we need other methods}$$
$$\text{In this case we cannot say anything and we need other tests}$$
$$\text{In this case we cannot proceed with standard methods}$$
And so on.
I got it, we need other methods. The question is What are those other methods? For I can't find something clear and understandable.
Maybe there are standard procedures like, I don't know: evaluate $f(x, y, \ldots)$ in some special points?
Often I read phrases like "from the graph you see..." or "coputing the graph you see that...". I don't think I can do the graph by pen and hand, when I am working only with pen and paper.
So my question is about good references, good examples, or anyway some good advice.
Thank you!
The Hessian matrix is a real symmetric matrix. Its eigenvalues are real, and if any two of those eigenvalues have opposite sign, the function is a saddle (not a local max or min). That much is familiar from the standard second derivative test. If all eigenvalues are say nonnegative, but some are zero, then it might be a local min but you need to confirm this using additional work. You need to explore the higher-order derivatives in the directions of the null eigenvectors. Call those the degenerate variables. Try finding the higher- order Taylor expansion of the function in these variables.
If you are lucky, there is only one degenerate variable, in which case you can see that if the first nontrivial term in its Taylor expansion is odd, the function changes sign (saddle). If it is even and negative like $- x^4$ it is again definitely a saddle. If it is even and positive then it is a local min.
When you have several degenerate variables you can generally try ranking the terms in the Taylor expansion by degree of homogeneity. Find the directions where the first term is (i) positive and (ii) where it is zero, then proceed to analyze the subcase (ii) by looking at the directional derivatives of higher order in the directions (ii). This can become tedious and tricky because the zero set of a multivariable polynomial can be geometrically quite complicated. There is an advanced sophisticated literature devoted to classification of singularities using $k$-jets (i,e. Taylor expansions) that is generally attributed to Rene Thom, and extended by others.