Suppose that we have a binary form $f(x,y) \in \mathbb{Z}[x,y]$ of degree 4, and that we explicitly have

$$\displaystyle f(x,y) = a_0 x^4 + a_1 x^3 y + a_2 x^2 y^2 + a_3 xy^3 + y^4,$$ so that $(0,1)$ is a solution to the equation $f(x,y) = 1$ (most notably, we require that the equation $f(x,y) = 1$ have a solution in $\mathbb{Z}^2$).

Suppose we define the auxiliary form $g = u_0 x^4 + \cdots + u_4 y^4$ by $u_0 = a_0$,

$u_1 = a_1 - 4 a_0 \left( \frac{b-c}{p^k}\right)$,

$u_2 = a_2 - 3a_1\left(\frac{b-c}{p^k}\right) + 6a_0 \left(\frac{b-c}{p^k}\right)^2$,

$u_3 = a_3 - 2a_2 \left(\frac{b-c}{p^k}\right) + 3a_1 \left(\frac{b-c}{p^k}\right)^2 - 4a_0 \left(\frac{b-c}{p^k}\right)^3$ and $u_4 = a_4 - a_3 \left(\frac{b-c}{p^k}\right) + a_2 \left(\frac{b-c}{p^k}\right)^2 - a_1 \left(\frac{b-c}{p^k}\right)^3 + a_0 \left(\frac{b-c}{p^k}\right)^4 $

where $p$ is a prime, $k \geq 1$ is an integer, and $b,c$ are such that each $u_i$ is an integer and $\gcd(u_0, \cdots, u_4) = 1$. Can one bound the number of choices of $b,c$ by an absolute constant for which $g(x,y) = 1$ also has a solution in $\mathbb{Z}^2$?

My motivation is the following situation. Stewart (my PHD advisor) conjectured in 1991 in "On the number of solutions of polynomial congruences and Thue equations", Journal of the American Mathematical Society, (4) Volume 4 (1991), 793-835 that there exists a constant $c_0$ such that for a given binary form $f(x,y) \in \mathbb{Z}[x,y]$ of degree $d \geq 3$, there exists a number $r(f) \in \mathbb{N}$ such that for all integers $h$ with $|h| \geq r(f)$, the number of *primitive* solutions to the equation $f(x,y) = h$ is bounded by $c_0$. A weaker form of the conjecture is to replace the number $c_0$ (which does not depend on the degree $d$) with a quantity $c(d)$ which depends on $d$. In the same paper, Stewart counted the solutions to $f(x,y) = h$ by considering a form $f(p^k x + by, y)$ with $p^k || h$ and where $b \pmod{p^k}$ is chosen so that the form $\tilde{f}(x,y) = f(p^k x + by, y)$ has content $p^k$. Then one obtains an equation of the form $f'(x,y) = h/p^k$, and continuing in this manner one eventually gets to a counting problem that one can solve. The above construction in my question 'reverses' this process, by first starting with a form $f(x,y)$ which manifestly has a solution to the equation $f(x,y) = 1$, then trying to lift the solution to one of the form $f'(x,y) = p^k$. My question is basically asking under what circumstances and two distinct forms (in my case, both degree 4) $f,g$ lift to the same form $f'$.

Edit 06 Sep 2014: Fixed the expressions for $u_i$, made calculation errors in the original.