Consider the bivariate polynomial $$p(X,Y) = X^5 - (2 Y + 1) X^3 - (Y^2 + 2) X^2 + Y (Y-1) X + Y^3.$$ For every integer $y \ge 4$, I conjecture that $p(X,y)$ is irreducible in $\mathbb{Q}[X]$. How can this be proved?

## Motivation

This infinite family of polynomials comes from a recent paper of mine. We wanted to show that the roots of every polynomial in this family satisfy what we call the lattice condition (Definition 6.3). For this family of polynomials, it is sufficient that they are each irreducible. However, we are unable to prove that they are in fact irreducible. Instead, with much effort, we find a different way to show that the roots of all these polynomials do indeed satisfy the lattice condition.

I am wondering if our proof could be simplified by proving the sufficient condition that these polynomials are irreducible.

## What I Know

- $p(X,y)$ has three distinct real roots and two nonreal complex conjugate roots (since the discriminant is negative).
- The algebraic curve defined by $p(X,Y)$ has genus 3 (computed by Maple).
- From Theorem 1.2 in Hilbert’s irreducibility theorem for prime degree and general polynomials by Peter Müller, it follows that $p(X,y)$ is reducible in $\mathbb{Q}[X]$ for at most a finite number of $y \in \mathbb{Z}$. (Note that this result is not effective.)
- The only values of $y \in \mathbb{Z}$ for which I know that $p(X,y)$ is reducible are $$ p(X,y) = \begin{cases} (X - 1) (X^4 + X^3 + 2 X^2 - X + 1) & y = -1\\ X^2 (X^3 - X - 2) & y = 0\\ (X + 1) (X^4 - X^3 - 2 X^2 - X + 1) & y = 1\\ (X - 1) (X^2 - X - 4) (X^2 + 2 X + 2) & y = 2\\ (X - 3) (X^4 + 3 X^3 + 2 X^2 - 5 X - 9) & y = 3. \end{cases}$$ These five reducible cases also give five integer solutions to $p(X,Y) = 0$.
- With help from Aaron Levin and Bjorn Poonen, we prove (Lemma 7.6) that the only integer solutions to $p(X,Y) = 0$ are the five solutions given by the five factorizations above. (The proof uses Puiseux series expansions.)
- Using Mathematica, I checked that $p(X,y)$ is irreducible for $4 \le y \le 2.1 \times 10^9$.

## Conclusion

From item 5, if $p(X,y)$ is reducible, then it must be a product of an irreducible quadratic and an irreducible cubic. Then using item 1 and some Galois theory, we prove (Lemma 7.7) that the roots of $p(X,y)$ satisfy the lattice condition in this case.

I am wondering if we can avoid this line of reasoning and instead prove that $p(X,y)$ is in fact irreducible in $\mathbb{Q}[X]$ for every integer $y \ge 4$.