I am given an exercise of my thesis which is about "Computational Algeberaic Geometry", but I dont have enough knowldege to do it or even think about it. The exercixe is:
A ternary cubic form is a homogeneous degree 3 polynomials in three variables and the space of ternary cubic has dimension 10. I need to find a basis of seven harmonic cubics which ones killed by the Laplace operator. In this way, we can also compute the dimension of the space of harmonic homogeneous polynomials of degree $d$ in $n+1$ variables, for each of the seven cubics $f$, we should find the seven eigenvectors, that is the vectors $v \in \mathbb{C}^3$, up to scalar, such that the gradient $(f_x, f_y, f_z)$ is proportional to $(x,y,z)$.
In some ways it is related to $3$-Veronese Embedding of $\mathbb{P}^2$. Any help, hint and guidance are greatly appreciated.
Let $\mathbb{P}_k^{9}$ parametrise the space of ternary cubics over $k$, where the $i^{th}$ coordinate gives the coefficient of the $i^{th}$ monomial in, say, lexigraphical order. The Laplacian is a linear operator which takes the space of ternary cubics to the space of homogeneous linear forms on $k$. That is, we may view $\nabla^2$ a linear map $$\nabla^2 : \mathbb{P}_k^{9} \to \mathbb{P}_k^{2}$$ on the parameter spaces. Say given by multiplying by a matrix $A \in \operatorname{Mat}_{3 \times 10}(k)$ (defined up to scalar multiplication by a unit).
The kernel of this matrix gives corresponds to the linear subspace of $\mathbb{P}_k^{9}$ parametrising ternary cubics vanishing under $\nabla^2$. Given the coefficient space it is obvious how to write down the original cubics.
If you are allowing rings it should be clear how to generalise this, as with higher degree polynomials. The above is also more or less an algorithm for computing this space too.