Critical points of polynomial functions

Khazhgali Kozhasov
Wednesday, March 14, 2018 - 16:15

From linear algebra we know that a generic real symmetric matrix $A=(a_{ij})_{i,j=1}^n$ has $n$ distinct real eigenvalues. A geometric reason for this is that the set of symmetric matrices with repeated eigenvalues has codimension two in the space of all symmetric matrices. Let's denote by $f_A(x)=x^tAx=\sum_{i,j=1}^n a_{ij}x_ix_j$ the quadratic form associated to the symmetric matrix $A=(a_{ij})$. Then critical points and critical values of $f_A: S^{n-1}\rightarrow \K{R}$ (the restriction of $f_A$ to the unit sphere) are exactly unit eigenvectors and eigenvalues of $A$. In particular, the number of critical points of $f_A|_{S^{n-1}}$ equals $2n$ for any generic $A$. If, now, $f=\sum_{i_1,\dots,i_d=1}^n a_{i_1\dots i_d}x_{i_1}\dots x_{i_d}$ is a homogeneous form of degree at least $d\geq 3$ the number $C(f)$ of critical point of $f|_{S^{n-1}}$ is not generically constant anymore. However, there is an upper bound on this number:\begin{align}C(f)\leq 2\frac{(d-1)^n-1}{d-2},\end{align} where $f$ is a generic degree $d\geq 1$ form in $n\geq 2$ variables.I will explain how to construct for any $d$ and $n$ generic forms attaining this bound. Moreover, I will show that the bound is attained by harmonic forms, known as spherical harmonics. The presentation will be as elementary as possible.

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