Conjecture 3.26

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $P\in\mathbb Z[X,Y]$ be irreducible in $\mathbb Q(X)[Y]$ and of positive $Y$-degree. Set $d=\deg P$ and $H(P)=\max|a_{ij}|$ for its integer coefficients. An admissible specialization is a positive integer $n$ for which $P(n,Y)$ is irreducible in $\mathbb Q[Y]$.

Statement recorded in the supplied source.

Is there a polynomial bound $B(d,H)$, uniform over these $P$, such that an admissible $n\le B(d,H(P))$ always exists? [S2]

Short English statement

Can one always find a positive irreducible specialization within a bound polynomial in the polynomial degree and coefficient height?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700044 (AMR-086-0044). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Conjecture 3.26 and preceding Hilbert irreducibility paragraph. https://arxiv.org/abs/math/0312440.