Authors’ abstract: “Let \(\mu\) be a probability measure on \(\mathbb{Z}\) that is not a Dirac mass and that has finite support. We prove that if the coefficients of a monic polynomial \(f(x) \in \mathbb{Z}[x]\) of degree \(n\) are chosen independently at random according to \(\mu\) while ensuring that \(f(0) \neq 0\), then there is a positive constant \(\theta = \theta (\mu)\) such that \(f(x)\) has no divisors of degree \(\leq \theta n\) with probability that tends to 1 as \(n \to \infty\).
Furthermore, in certain cases, we show that a random polynomial \(f(x)\) with \(f(0) \neq 0\) is irreducible with probability tending to 1 as \(n \to \infty\). In particular, this is the case if \(\mu\) is the uniform measure on a set of at least 35 consecutive integers, or on a subset of \([-H,H] \cap \mathbb{Z}\) of cardinality \(\geq H^{4/5}(\log H)^2\) with \(H\) sufficiently large. In addition, in all of these settings, we show that the Galois group of \(f(x)\) is either \(\mathcal{A}_n\) or \(\mathcal{S}_n\) with high probability.
Finally, when \(\mu\) is the uniform measure on a finite arithmetic progression of at least two elements, we prove a random polynomial \(f(x)\) as above is irreducible with probability \(\geq \delta\) for some constant \(\delta =\delta (\mu)>0\). In fact, if the arithmetic progression has step 1, we prove the stronger result that the Galois group of \(f(x)\) is \(\mathcal{A}_n\) or \(\mathcal{S}_n\) with probability \(\geq \delta\).”
Reviewer’s comments: This eighty-pages-long paper is devided in four parts (besides the acknowledgements and the references). In Part I one finds the statements of the results and the outline of the proofs thereof (already thirty-one pages on the whole). Part II (ten pages) contains known and new results on approximate distribution, while Part III (twenty-one pages) handles many criteria of sets of polynomials, in comparison to their irreducibility or not. In Part IV, corresponding structure of the Galois groups, it is discussed with emphasis the so-called Łuczak-Pyber style theorem (twelve pages).
The paper does contain an immense lot of knowledge, very recent and not-so-very recent; moreover the reader will observe, that the printing of each page of the paper does contain about \(35\sim 40\) lines besides the normal “white spaces”! Now, let us give the titles of the subdivisions of the parts, in order to provide an idea of the contents.
I: Introduction; Outline of the proofs (Ruling out factors of small degree; Ruling out factors of large degree; From approximate equidistribution to irreducibility; Controlling the joint distribution of \((A_p)_{p\in P}\); A master theorem; From irreducibility to Galois groups; Summary (reviewer’s remark: here the authors print a “Leitfaden”, how the propositions and theorems do depend on each other)); Deduction of Theorems 1 to 6 from Theorems 7 and 8 (reviewer’s remark: It takes fifteen pages to do so, directly yielding the hart of the paper).
II: The Fourier transform on \(\mathbb{F}_P[T]\); \(L^\infty\) bounds; \(L^1\) bounds; Proof of Proposition 2.3.
III: Ruling out factors of small degree; An upper bound sieve; Anatomy of polynomials (thirteen pages in all, perhaps the most intricate and complex mathematics in its presentation (reviewer’s remark), highly recommended as such); Proof of Proposition 2.2.
IV: Galois theory (The factorization type of \(A_p\); Lifting the Frobenius automorphism; Reduction of Proposition 2.4 to two lemmas. A Łuczak-Pyber style theorem (The anatomy of a typical partition; Group theory)).
Let me finish with the following observations. The authors did succeed very well in presenting all the complications in the mathematics, in an excellent technical and didactic way. It must have costed them years, I guess, to conceive all that, let alone the typing of it.