I am studying proper holomorphic maps: one result is that there are no proper holomorphic maps from $B_n$ to $\Delta^n$, i.e. from the open ball in $\mathbb C^n$ in the open polydisc. What if I increase the dimension on the right?
I can't come to an answer: for what I know, embedding theorems could hold, such that every ball can be embedded holomorphically into some polydisc in great dimension.
Thank you in advance for any suggestion.
Yes, every ball can be holomorphically embedded into some polydisc of sufficiently high dimension.
More generally, Løw has proved that every strictly psudoconvex open subset $\Omega \subset \mathbb C^n$ with $C^2$ boundary can be properly holomorphically embedded into a polydisc $P\subset \mathbb C^m$ of sufficiently high dimension $m\gt n$.
This result is Corollary 1, page 453 of Løw's Theorem 1 in this 1985 Mathematische Zeitschrift article.
A complement
Although it was not asked in the question let me mention that the result does not hold in the other direction:
If $m\gt1$ and $r\gt0$ there is no holomorphic proper map of a polydisc $P \subset \mathbb C^m$ into an open ball $B\subset \mathbb C^r.$
This is part of Theorem 15.2.4, pages 306-307, of Rudins Function Theory in the Unit Ball of $\mathbb C ^n$.