Consider given integers $A,B$ such that $AB \neq 0$.
Consider a given polynomial $f(x) = a_0 + a_1 x + a_2 x^2 + ... $ of degree $n > 1$ with rational coefficients $a_i$.
Now I wonder about solving the diophantine equations of type :
$$ A(x+y_1)(x+y_2)...(x+y_n) + B(x+z_1)(x+z_2)...(x+z_k) = f(x) $$
where $y_i,z_i$ are fractions and $0<k<n$ is an integer.
In particular the cases when $k = 1,2,n$.
Also the similar equation
$$ A(x+w_1)(x+w_2)...(x+w_n) + C = f(x) $$
where $w_n$ are fractions and $C$ is a nonzero integer.
Already for small $n$ and small $A,B$ i got stuck trying to solve those.
Does vieta jumping help here ?
Is this a hard problem or an easy one ?
When do we have solutions ? Can we classify when we have solutions ?
Is this rather algebra or rather number theory ?
Can we parametrise the solutions for $n<5$ ?
EDIT:
For the case of the second type equation notice that if $C$ is an integer there are only finitely many possibilities for it. There exist upper bounds and lower bounds for $C$ due to the shape of the polynomial. We can combine that idea by considering all potential $C$ with the rational root test and factoring to make a list of all candidates $w_i$.
Therefore we have made a method to solve it because we made it computable, however this is very inefficient ofcourse.
The harder question of rational $C$ comes to mind. Maybe it reduces too. Probably.
The main question (equation 1) does not seem aided by this though.
Just some comments...