I want to write down a curve, which I define as the set of points $x,y\in\mathbb{F}$ (a field) such that $f(x,y)=0$, but I write it as the polynomial $f$ and call $f$ the curve. So I would like to write down a curve of degree $n$ in a formal way, like: \begin{equation*} f(x,y)=\sum_{i=0}^{n}\alpha_i x^i + \sum_{i=1}^{n-1}\beta_{i} x^i y^{n-i} + \sum_{i=0}^{n}\gamma_i y^i \end{equation*} where $\alpha_i, \beta_i, \gamma_i$ are the function's coefficients in $\mathbb{F}$, the field over which we defined $f$, and the summations should be read as taking all possible cross terms of $x$ and $y$.
My main question is: is it possible to write these three summations in a shorter way such that it describes all possible cross terms (all combinations of powers) and all powers of $x$ and $y$ in a formal way? I.e. without explaining what the sums mean, but it has to speak for itself. Thanks in advance for the help.
If you want all degree-$n$ terms as well as lower-order terms: $$f(x,y) = \sum_{\substack{i+j\le n \\ i,j\ge 0}}\alpha_{i,j} \;x^i y^j.$$ If you want it to be homogeneous of degree $n$: $$f(x,y) = \sum_{\substack{i+j = n \\ i,j\ge 0}}\alpha_{i,j} \;x^i y^j.$$ (If you want the TeX commands for these, right-click them, then "Show Math As" -> "TeX Commands".)