Let's say I have a multiset of complex numbers $\lbrace a\_1,\cdots,a\_n\rbrace$ (so some of the elements may be repeated) and I would like to construct an entire function $p(z)$ with those numbers as zeroes. However, I also have a multiset of complex numbers $B = \lbrace b\_1,\cdots,b\_n \rbrace$ such that I wish $p(b\_i) = 1$ - **p is only 1 on the $b\_i$'s**. 


It seems like trying to use Lagrange's polynomial interpolation formula gives you a polynomial with too high a degree (greater than $n$ and less than or equal to $2n$), and then there's the possibility that $p^{-1}(1) \nsubseteq B$. 


I've been thinking about doing the following:


Let $g(z) = (x-a\_1) \cdots (x - a\_n)$, and then via Weierstrass construct an entire function $h(z)$ such that $e^{h(b\_i)} = 1/g(b\_i)$. Then it seems like the entire function $e^{h(z)}g(z)$ is getting somewhat closer to what I want - but then again I don't know if there are any other $\alpha$'s such that $e^{h(\alpha)}g(\alpha) = 1$ where $\alpha \notin B$. 


The problem of polynomial interpolation and fitting seems very well studied; however, I can't seem to find a reference for this particular puzzle.


Thanks in advance!



Entire function interpolation with control over multiplicities/derivatives
If I read you right,
you want an entire function that takes the values $0$ and $1$ at only
finitely many (specified) points. This implies that the function must be a polynomial,
by Picard's great theorem, since there will be deleted neighbourhoods of
infinity where the function misses two values.


