Let $f_1,\ldots,f_r\in\mathbb{C}[x,y]$ and consider the subalgebras $A_1,\ldots,A_r$ of $\mathbb{C}[x,y]$ that are generated by $f_1,\ldots,f_r$, i.e., $A_i=\mathbb{C}[f_i]$. Using some dimension estimates, one can show that the $A_1,\ldots,A_r$ do not span $\mathbb{C}[x,y]$ as a vector space. Thus there is a nonzero linear form $L:\mathbb{C}[x,y]\to\mathbb{C}$ that vanishes on each $A_i$, or equivalently on all $f_i^m$. I am interested in whether we can find such an $L$ which can be written as a linear combination of point evaluations. Is this always the case? Are there some criteria on the $f_1,\ldots,f_r$ that guarantee this? Has this been studied somewhere?
For example if $r=2$, $f_1=x$ and $f_2=y$, then we choose $L=ev_{(0,0)}-ev_{(1,0)}-ev_{(0,1)}+ev_{(1,1)}$. Here $ev_p:\mathbb{C}[x,y]\to\mathbb{C}$ denotes the point evaluation at the point $p$.
