Hi. Is it possible to solve (some, all, no) bivariate polynomial functions of infinite degree?

I mean is there a theorem that says we can or cannot solve such an equation? I've tried looking at the Galois thing and Abel-Ruffian theorem on wikipedia and it's just totally Chinese to me.

Taylor power series are sort of like polynomial functions of infinite degree, granted, but they are not bivariate and the sequence equals the function only within the finite radius of convergence.

Thanks

Oh, PS on that:

I understand that there is a theorem that states that polynomial functions of degree n will have n roots. Does this mean that the infinite degree polynomial function I am asking about above would have infinite roots?