I have a degree-96 even irreducible polynomial $P(x)$ with integer coefficients between 90 and 110 decimal digits long. I would like to find a value of $x$ such that $P(x)$ is an integer square. (Of course the constant term is not a square.)
Is there a technique or algorithm that would let me find such an $x$, or rule out that such an $x$ exists? (I can't find a modular obstruction.) Of course it need not be specific to my case; I list my parameters to give an idea of the scale of my problem.
There is a question asking the same thing about the quadratic case, but this is far harder. In particular, with degree $\ge5$ Galois theory suggests this will be hairy. :-)