The characteristic polynomial of a real symmetric $n\times n$ matrix $H$ has $n$ real roots, counted with multiplicity. Therefore the discriminant $D(H)$ of this polynomial is zero or positive. It is zero if and only if there is a degenerate eigenvalue.
Thus $D(H)$ is a non-negative (homogeneous) polynomial in the $\frac12n(n+1)$ entries of $H$. Some non-negative polynomials can be written as a sum of squares and I am interested in whether $D(H)$ can. There is a concrete question at the end, but any insights into the general case are also welcome.
The size of the problem grows very quickly with dimension, so I will only look at $n=2$ (which I do understand) and $n=3$ (which I am yet to understand).
2D
In two dimensions it is pretty easy to write down the polynomial and its discriminant and see by eye that $$ D(H) = (h_{11}-h_{22})^2 + 4h_{12}^2, $$ which is indeed a sum of two squares.
Having a degenerate eigenvalue is a polynomial condition: it happens if and only if $D(H)=0$. The discriminant is a second order polynomial, but writing it as a sum of squares leads to far simpler algebraic condition: $h_{11}-h_{22}=0$ and $h_{12}=0$. Simple algebraic conditions for degeneracy are the goal here, but I thought the question would be of some interest in itself.
3D
In three dimensions the discriminant is pretty big: $$ D(H) = h_{22}^2h_{33}^4-2h_{11}h_{22}h_{33}^4+4h_{12}^2h_{33}^4+h_{11}^2h_{33}^4-2h_{22}h_{23}^2h_{33}^3+2h_{11}h_{23}^2h_{33}^3-8h_{12}h_{13}h_{23}h_{33}^3-2h_{22}^3h_{33}^3+2h_{11}h_{22}^2h_{33}^3+2h_{13}^2h_{22}h_{33}^3-8h_{12}^2h_{22}h_{33}^3+2h_{11}^2h_{22}h_{33}^3-2h_{11}h_{13}^2h_{33}^3-8h_{11}h_{12}^2h_{33}^3-2h_{11}^3h_{33}^3+h_{23}^4h_{33}^2+8h_{22}^2h_{23}^2h_{33}^2-10h_{11}h_{22}h_{23}^2h_{33}^2+2h_{13}^2h_{23}^2h_{33}^2+20h_{12}^2h_{23}^2h_{33}^2+2h_{11}^2h_{23}^2h_{33}^2+12h_{12}h_{13}h_{22}h_{23}h_{33}^2+12h_{11}h_{12}h_{13}h_{23}h_{33}^2+h_{22}^4h_{33}^2+2h_{11}h_{22}^3h_{33}^2+2h_{13}^2h_{22}^2h_{33}^2+2h_{12}^2h_{22}^2h_{33}^2-6h_{11}^2h_{22}^2h_{33}^2-10h_{11}h_{13}^2h_{22}h_{33}^2+20h_{11}h_{12}^2h_{22}h_{33}^2+2h_{11}^3h_{22}h_{33}^2+h_{13}^4h_{33}^2+20h_{12}^2h_{13}^2h_{33}^2+8h_{11}^2h_{13}^2h_{33}^2-8h_{12}^4h_{33}^2+2h_{11}^2h_{12}^2h_{33}^2+h_{11}^4h_{33}^2-10h_{22}h_{23}^4h_{33}+8h_{11}h_{23}^4h_{33}-36h_{12}h_{13}h_{23}^3h_{33}-2h_{22}^3h_{23}^2h_{33}-10h_{11}h_{22}^2h_{23}^2h_{33}-2h_{13}^2h_{22}h_{23}^2h_{33}-2h_{12}^2h_{22}h_{23}^2h_{33}+20h_{11}^2h_{22}h_{23}^2h_{33}-2h_{11}h_{13}^2h_{23}^2h_{33}-38h_{11}h_{12}^2h_{23}^2h_{33}-8h_{11}^3h_{23}^2h_{33}+12h_{12}h_{13}h_{22}^2h_{23}h_{33}-48h_{11}h_{12}h_{13}h_{22}h_{23}h_{33}-36h_{12}h_{13}^3h_{23}h_{33}+72h_{12}^3h_{13}h_{23}h_{33}+12h_{11}^2h_{12}h_{13}h_{23}h_{33}-2h_{11}h_{22}^4h_{33}-8h_{13}^2h_{22}^3h_{33}+2h_{12}^2h_{22}^3h_{33}+2h_{11}^2h_{22}^3h_{33}+20h_{11}h_{13}^2h_{22}^2h_{33}-10h_{11}h_{12}^2h_{22}^2h_{33}+2h_{11}^3h_{22}^2h_{33}+8h_{13}^4h_{22}h_{33}-38h_{12}^2h_{13}^2h_{22}h_{33}-10h_{11}^2h_{13}^2h_{22}h_{33}+8h_{12}^4h_{22}h_{33}-10h_{11}^2h_{12}^2h_{22}h_{33}-2h_{11}^4h_{22}h_{33}-10h_{11}h_{13}^4h_{33}-2h_{11}h_{12}^2h_{13}^2h_{33}-2h_{11}^3h_{13}^2h_{33}+8h_{11}h_{12}^4h_{33}+2h_{11}^3h_{12}^2h_{33}+4h_{23}^6+h_{22}^2h_{23}^4+8h_{11}h_{22}h_{23}^4+12h_{13}^2h_{23}^4+12h_{12}^2h_{23}^4-8h_{11}^2h_{23}^4-36h_{12}h_{13}h_{22}h_{23}^3+72h_{11}h_{12}h_{13}h_{23}^3+2h_{11}h_{22}^3h_{23}^2+20h_{13}^2h_{22}^2h_{23}^2+2h_{12}^2h_{22}^2h_{23}^2+2h_{11}^2h_{22}^2h_{23}^2-38h_{11}h_{13}^2h_{22}h_{23}^2-2h_{11}h_{12}^2h_{22}h_{23}^2-8h_{11}^3h_{22}h_{23}^2+12h_{13}^4h_{23}^2-84h_{12}^2h_{13}^2h_{23}^2+20h_{11}^2h_{13}^2h_{23}^2+12h_{12}^4h_{23}^2+20h_{11}^2h_{12}^2h_{23}^2+4h_{11}^4h_{23}^2-8h_{12}h_{13}h_{22}^3h_{23}+12h_{11}h_{12}h_{13}h_{22}^2h_{23}+72h_{12}h_{13}^3h_{22}h_{23}-36h_{12}^3h_{13}h_{22}h_{23}+12h_{11}^2h_{12}h_{13}h_{22}h_{23}-36h_{11}h_{12}h_{13}^3h_{23}-36h_{11}h_{12}^3h_{13}h_{23}-8h_{11}^3h_{12}h_{13}h_{23}+4h_{13}^2h_{22}^4+h_{11}^2h_{22}^4-8h_{11}h_{13}^2h_{22}^3-2h_{11}h_{12}^2h_{22}^3-2h_{11}^3h_{22}^3-8h_{13}^4h_{22}^2+20h_{12}^2h_{13}^2h_{22}^2+2h_{11}^2h_{13}^2h_{22}^2+h_{12}^4h_{22}^2+8h_{11}^2h_{12}^2h_{22}^2+h_{11}^4h_{22}^2+8h_{11}h_{13}^4h_{22}-2h_{11}h_{12}^2h_{13}^2h_{22}+2h_{11}^3h_{13}^2h_{22}-10h_{11}h_{12}^4h_{22}-2h_{11}^3h_{12}^2h_{22}+4h_{13}^6+12h_{12}^2h_{13}^4+h_{11}^2h_{13}^4+12h_{12}^4h_{13}^2+2h_{11}^2h_{12}^2h_{13}^2+4h_{12}^6+h_{11}^2h_{12}^4 . $$ (I got this by Maxima.) This is indeed a non-negative homogeneous polynomial of degree six in six variables, but it is too big for me to see any structure by eye and I cannot tell whether it is a sum of squares.
In the diagonal case $h_{12}=h_{13}=h_{23}=0$ the discriminant has a simpler expression: $$ D(H) = (h_{11}-h_{22})^2 (h_{22}-h_{33})^2 (h_{11}-h_{33})^2. $$ This form is not at all surprising, as it should be a sixth degree polynomial vanishing if and only if two diagonal entries coincide.
My concrete question is: Is this $D(H)$ of the case $n=3$ a sum of squares (without assuming it is diagonal)? If yes, what are the squared polynomials and how unique are they?
I have understood that there are computational tools for finding a sum of squares decomposition, but I have yet to find one that I could run with the software I have. And I assume this particular polynomial has structure which simplifies matters: for example, the polynomial is invariant under orthogonal changes of basis and the non-negativity has a geometric meaning. One can indeed diagonalize the matrix, but I cannot see a way to use this to understand what the polynomial is in terms of the original basis. The 2D case and the diagonal 3D case suggest that being a sum of squares is a reasonable guess.
