I have a linear operator over the complex field that is defined as the second derivative of x and y $$ \frac{\partial}{\partial x^2} + \frac{\partial}{\partial y^2} $$ V is the vector space of all polynomials of the variables x and y with degree max n with complex coefficients. I want to find the index of this operator. I know that the degree will be less than the dimension of the operator, but it seems like their should be a lower upper bound that is somehow related to the "middle" exponents of the binomial expansion of (x+y)^n (e.g. the operator is nilpotent with an index of max 2 for polynomials of degree of 4). But I don't know how to describe this in a better way. Any help would be appreciated
Edit: No response yet. I am of course using the basis of x and y. For example for polynomials of degree max 2, I use the basis (1,x,y,x^2,xy,y^2), but I can't seem to figure out how to advance