What does it mean for a member of formal power series over a field to be algebraic over polynomial ring of that field?

Question

What does it mean for a member of formal power series over a field to be algebraic over polynomial ring of that field? For example what does it mean for a $f$ in $k[[t_1 ,...,t_n ]]$ which is algebraic over $k[t_1 ,...,t_n]$ ?

Answer 1

It means that $f$ satisfies a polynomial equation with coefficients in $k[t_1 ,...,t_n]$.

For instance, $f(t) = 1 + t + t^2 + t^3 + \cdots$ satisfies $(1-t)f-1=0$, a polynomial equation with coefficients in $k[t]$. In other words, $f$ is a root of $(1-t)X-1 \in k[t][X]$.