nt.number theory - Condition for existence of certain lattice points on polytopes

Let $a_1,cdots, a_n$ be integers such that $a_igeq 2$ for all $i$ and $k>0$ another integer.

I am interested in whether there exist integers $x_1,cdots, x_n$ with $0<x_i<a_i$ satisfying:

$$ frac{x_1}{a_1}+ cdots + frac{x_n}{a_n}=k (*)$$

For example, if $n=2,k=1$, there exist a solution in the specified range iff $text{gcd}(a_1,a_2) neq 1$, in other words $text{lcm}(a_1,a_2)-a_1a_2 neq 0$.

Question: For what pairs $n,k$ is there a polynomial $F$ whose inputs are $text{lcm}$ of various subsets of ${a_i}_{1leq ileq n}$ such that (*) has a solution in the specified range if and only if $Fneq 0$ (see some examples I have in mind below)?

Some observations: If $kgeq n$ then $Fequiv 0$ works, trivially.

Some calculations for small values of $(n,k)$ suggest the following strange function: for each subset $I subset {1,cdots,n}$ define
$$f_I(a_1,cdots,a_n):= frac{prod_{iin I}a_i}{text{lcm}{a_i}_{iin I}}$$

For $I=varnothing$ we set $f_I=1$. Let:

$$F(a_1,cdots,a_n):= sum_{Isubset {1,cdots,n}}(-1)^{|I|}f_I(a_1,cdots,a_n) $$

(You can replace $F$ by a polynomial of the $text{lcm}$s which vanishes at the same time). This works for $(n,k)=(2,1)$ and probably $(3,1)$ (which would imply $(3,2)$). It does not work for $(4,1)$ but seems to work for some sequence with $(n,k)= (4,2)$ (may be the function depends on $n/k$?). Have anyone seen this kind of formula before in other contexts?

PS: I am not sure what tags should be used. Please feel free to re-tag.

EDIT: Apparently this question is related to existence of lattice points on a polytope. I checked through some of the references pointed to in this question on MO, but could not find the exact answer to what I wanted.