Let K be a field and f(x), g(x) ∈ K[x]. It is trivial that f(x)− g(y) has a linear factor if and only if f(x) = g(ax+ b), where a ∈ K∗ and b ∈ K. The problem when f(x)− g(y) has a quadratic factor is considerably more complicated. If f = φ ◦ f1 and g = φ ◦ g1, where φ(x), f1(x), g1(x) ∈ K[x] and max(deg f1,deg g1) = 2 then, trivially, f(x)− g(y) has the quadratic factor f1(x)− g1(y). However, there also is a famous series of nontrivial examples, provided by the Chebyshev polynomials: Tn(x) + Tn(y) splits (over an algebraically closed field) into quadratic factors (and one linear factor if n is odd; see Proposition 3.1). Recall that the Chebyshev polynomials are defined from Tn(cosx) = cosnx, or, alternatively, from Tn((z + z−1)/2) = (z + z−n)/2. In this note we completely solve Problem 1.1 for polynomials over a field of characteristic 0. We start from the case of algebraically close base field, which is technically simpler.

/pdf/quadratic-factors-of-f-x-g-y-3k7sybzr44.pdf

Quadratic factors of f(x) -g(y)

In this article, we give several conditions implying the irreducibility of the algebraic curve P(x) − Q(y) = 0, where P, Q are rational functions. We also apply the results obtained to the functional equations P(f) = Q(g) and P(f) = cP(g), where c ∈ ℂ. For example, we show that for a generic pair of rational functions P, Q the first equation has no non-constant solutions f, g meromorphic on ℂ whenever (deg P − 1)(deg Q − 1) ≥ 2.

Algebraic curves P(x) − Q(y) = 0 and functional equations

In this paper we give several conditions implying the irreducibility of the algebraic curve P(x)-Q(y)=0, where P,Q are rational functions. We also apply the results obtained to the functional equations P(f)=Q(g) and P(f)=cP(g), where c\in C. For example, we show that for a generic pair of rational functions P,Q the first equation has no non-constant solutions f,g meromorphic on C whenever (°P-1)(°Q-1) \geq 2.

Algebraic curves P(x)-Q(y)=0 and functional equations

A pair of separable polynomials (g, h) over a finite field F is said to be a strong Davenport pair if g(K) = h(K) for all finite extension fields K/F , where g(K) = { g(a) | a ∈ K }. We construct examples for which g is a projective or additive polynomial, and we find a factorization of g(x)−h(y). In addition, we prove that |g−1(a)∩K| = |h−1(a) ∩ K| for all a ∈ K. Though the existence of such examples was known to Fried, the explicit formulas for g and h are new, as are our methods of proof. This article also contains new results about additive and projective polynomials. 2000 Mathematics Subject Classification: 12E10, 12E05

/pdf/explicit-formulas-for-strong-davenport-pairs-40jpcdj36x.pdf

Explicit formulas for strong Davenport pairs

Dans toute la suite K désigne un corps commutatif de caractéristique nulle, K une clôture algébrique de K et X = (X1, . . . ,Xn) (n ≥ 2) des indéterminées algébriquement indépendantes sur K, que nous noterons X et Y pour n = 2. Cet article, dans la suite de travaux de Bertini, Krull et Stein, a pour objet de montrer que pour tout ensemble {a1, . . . , as} d’éléments distincts de K et pour tous entiers %1, . . . , %s positifs non nuls, il existe un polynôme P (X) dans K[X] non composé tel que P (X) − ai ait exactement %i + 1 facteurs irréductibles dans K[X], i = 1, . . . , s, et P (X)− λ soit irréductible pour tout λ 6∈ {a1, . . . , as}.

Sur le spectre d'un polynome `a plusieurs variables

The monodromy method—featuring braid group action—first appeared as a moduli space approach for finding solutions of arithmetic problems that produce reducible variables separated curves. Examples in this paper illustrate its most interesting aspect: investigating the moduli space of exceptions to a specific diophantine outcome. Explicit versions of Hilbert’s irreducibility theorem and Davenport’s problem fostered this technique and motivated the genus 0 problem started by J. Thompson and taken up by many group theorists. We review progress on the genus 0 problem in 0 characteristic, and its quite different contributions in positive characteristic. Example: Let f and h be polynomials with coefficients in a number field K. The classification of finite simple groups shows there is a bound on exceptional degrees for f to the following result. If f is indecomposable and h is not a composition with f , then f(x) − h(y) is irreducible. This answered challenge problems on factorization of variables separated polynomials posed by A. Schinzel in the early 60’s. This limitation result holds, however, only in characteristic 0, one difference between the Genus 0 Problem here and in positive characteristic. Finite field example—Davenport’s Problem: For each finite field Fq there are infinitely many surprising polynomial pairs (f, h) (of degree prime to the characteristic) whose value sets are equal over Fqt , t = 1, 2, . . . . Though we include unpublished results from 30 years ago, a new set of problems stretch the methods. Example of a general theme: Let Md be the elements of Q of degree no more than d over Q. The degree d reducibility set of f , Rf (d) is Rf (d) = {z0 ∈ Md | f(y) − z0 is reducible over Q(z0)}. Similarly, there is a value set Vf (d) (degree 1 fiber). Following a reduction due to G. Frey, for any integer d, there are polynomials f of unbounded degree satisfying Rf (d) \ Vf (d) is finite. The full monodromy method finds precise arithmetic information by extending observations from modular curves. A semi-classical observation: Divisors with support in cusp points on modular curves generate a torsion group on the Jacobian of the curve. Illustrations here (from alternating group covers) show generalizations of this issue are ubiquitous with the monodromy method. Date: October 25, 2001. 1991 Mathematics Subject Classification. Primary 12F05; Secondary 14H40, 14D20, 14E20.

Variables separated polynomials, the genus 0 problem and moduli spaces

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Variables separated polynomials, the genus 0 problem and moduli spaces