Van der Poorten conjectures [in "Power series representing algebraic functions," *Sem. Th. Nombres Paris 1990-91*] that if a power series over the rationals is the [complete] diagonal [of a *rational* power series in several variables], then it is algebraic over $\mathbb{Q}(X)$ iff it is algebraic [modulo almost every prime] $p$, [of degree] bounded independiently of $p$.

I want to know if there are [any] results [available in this direction].

[*Remark*: Not that I know of, except of course that the diagonal of a rational power series in *two* variables is always algebraic. -- V.D.]