Filaseta 2000 irreducibility theorem

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corollary_p3: Filaseta, Ford and Konyagin's corollary that for coprime f and g in Z[x] with nonzero constant terms and n at least an explicit bound exponential in N = 2||f||^2 + 2||g||^2 + 2r_1 + 2r_2 - 7, the non-reciprocal part of f(x)x^n + g(x) is irreducible or identically 1 or -1, except when minus fg is a pth power for a prime p dividing n or, for a common sign e = 1 or -1, one of ef and eg is a fourth power and the other four times a fourth power, with 4 | n.

illinois_talk_1999: Identifies and compares the authors' lecture-slide version with the article-manuscript corollary on lacunary-polynomial irreducibility.

theorem_1: Filaseta, Ford and Konyagin's theorem that if F in Z[x] has degree above a doubly exponential bound in N = 2||F||^2 + 2r - 5 and its non-reciprocal part is reducible, then for some integer k in [k_0, deg F] the polynomial obtained by splitting each exponent of F as a multiple of k plus its residue mod k, with y standing for x^k, is reducible in Z[x,y].

theorem_2: Filaseta, Ford and Konyagin's refinement of their Theorem 1: if deg F is at least max{2 x 5^(2N-1), k_0(5^(N-1) + 1/4)} with N = 2||F||^2 + 2r - 5 and the non-reciprocal part of F is reducible, then for some integer k in [k_0, 4(deg F)/3) every exponent of F lies less than k/4 from a multiple of k, and the lift of x^[k/4]F, with the largest power of x removed, is reducible in Z[x,y].

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Filaseta, M. and Ford, K. and Konyagin, S., On an irreducibility theorem of {A}. Schinzel associated with coverings of the integers. Illinois J. Math. 44 (2000), no. 3, 633--643. DOI 10.1215/ijm/1256060421.

Motivated largely by Schinzel's work linking the reducibility of f(x)x^n + 1 to coverings of the integers, and through it by the odd-covering problem on which Erdos and Selfridge bet, the paper reworks Schinzel's theorem on the non-reciprocal part of lacunary polynomials with explicit bounds. Theorem 1 (p. 2) shows that if F(x) = sum a_j x^{d_j}, with 0 = d_0 < ... < d_r and all a_j nonzero, has degree at least a doubly exponential bound in N = 2||F||^2 + 2r - 5 and its non-reciprocal part is reducible in Z[x], then for some integer k in [k_0, deg F] the two-variable polynomial G(x,y) = sum a_j x^{d_j mod k} y^{l_j} is reducible in Z[x,y]; Theorem 2 (p. 3) is a refinement, inserting a shift by [k/4] and removing a power of x, which replaces the doubly exponential bound by an exponential one, takes k in [k_0, 4(deg F)/3), and forces at least one exponent of y to be positive so that the conclusion does not follow at once from the reducibility of F. The unnumbered Corollary (p. 3) applies this to f(x)x^n + g(x) with f, g coprime and nonzero constant terms: for n above an explicit bound the non-reciprocal part is irreducible or identically 1 or -1 unless -f(x)g(x) is a p-th power for some prime p dividing n, or, for one common sign e = +/-1, one of e f, e g is a fourth power and the other 4 times a fourth power with 4 | n; the paper credits the case f = 1, without an explicit bound, to Schinzel. The method ties reducibility of non-reciprocal parts to an elementary problem about the residues of the exponents d_j modulo k, treated in Section 2. The paper recalls (p. 1) Schinzel's result that a polynomial f with f(1) not equal to -1 and f(x)x^n + 1 reducible for every positive integer n would force an odd covering of the integers, and says (p. 2) that its approach gives factorization information on f(x)x^n + 1 sufficient to carry out that connection.

Source: https://ford126.web.illinois.edu/papers-ann.html.

Versions. The copy read for this card is the ten-page article manuscript, 200,452 bytes, internally numbered 1--10 and corresponding to the work published in Illinois Journal of Mathematics 44 (2000), 633--643; it carries no journal-facsimile header, so publisher-facsimile identity has not been established. The talk slides, 150,425 bytes, are the 20-page slides of the authors' invited lecture at the AMS Sectional Meeting in Urbana-Champaign on 20 March 1999. The slide theorem, headed "Theorem (F., Ford, Konyagin)", is an abbreviated form of the manuscript Corollary; see [[covering_systems/filaseta_2000_irreducibility_theorem/illinois_talk_1999|the statement comparison]]. The article manuscript PDF prints no copyright or license line, and the author's publication list that provides it states no terms (https://ford126.web.illinois.edu/papers-ann.html, read 2026-10-02); the term is unstated. The talk slides PDF likewise prints no copyright or license line and is provided by the same list; the term is unstated.

Read status. Claims checked: Theorems 1 and 2, the Corollary and the remarks after them were read clause by clause on the manuscript's page images. The proofs (pp. 4--10) were read but not checked step by step.

Bears on. #7: the Corollary with g = 1 gives, for n beyond an explicit bound, the irreducibility information on the non-reciprocal part of f(x)x^n + 1 that the paper says suffices for Schinzel's link between such polynomials and odd coverings. The paper constructs no covering and does not decide whether an odd covering exists.

Results. Theorem 1 (p. 2); Theorem 2 (p. 3); Corollary (p. 3, unnumbered). Section 2 (pp. 4--8) proves the residue lemmas behind the theorems: Lemma 1 (p. 4) finds, for an integer r >= 2, numbers 1 = x_1 > x_2

... > x_r in [0,1] and alpha in (0,1/2], a real b in [1, B_r(alpha)] with every fractional part {b x_j} below alpha; Lemma 2 (p. 5) finds, for real k_0 >= 2 and non-negative integers a_1 < ... < a_r with a_r at least a doubly exponential A(r), an integer k in [k_0, a_r] with every a_j mod k below k/2; Lemma 3 (p. 6) finds, once a_r is at least an exponential A'(r), an integer k in [k_0, 4a_r/3) with every a_j mod k in [0,k/4) or (3k/4,k). Examples 1 and 2 (pp. 7--8) show that A(r) must grow doubly exponentially and A'(r) exponentially. These are proof steps, summarized here and given no pages of their own.

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