Polynomials with a common composite

Abstract

Abstract. Let f and g be nonconstant polynomials over a field K. In this paper we study the pairs (f, g) for which the intersection K[f ]∩K[g] is larger than K. We describe all such pairs in case K has characteristic zero, as a consequence of classical results due to Ritt. For fields K of positive characteristic we present various results, examples, and algorithms.

Decomposable polynomials in second order linear recurrence sequences

“…), then f 1 and f 2 are said to have a common composite. 'Most' pairs of complex polynomials have no common composite (this follows to the most part already from Ritt's results, see [1] for the details).…”

“…To detect cases when there does not exist a vanishing subsum of (4), we apply several tools. We follow a Galois-theoretic approach to decomposition questions, which originated in Ritt's work [21], and apply some recent results on polynomial decomposition from [1] and [20]. We show that the following holds.…”

“…. , n. We say that the recurrence relation (1) is minimal if (G n ) ∞ n=0 does not satisfy a recurrence relation with smaller d and coefficients in C[x]. We say that (1) is non-degenerate if α i /α j ∈ C * for all i = j.…”

Two Questions on Polynomial Decomposition

“…If f (x) ∈ R[x] has degree at least 2, we say that f is decomposable (over R) if we can write f (x) = g(h(x)) for some nonlinear g, h ∈ R[x]; otherwise we say f is indecomposable. Many authors have studied decomposability of polynomials in case R is a field (see, for instance, [1,2,4,5,8,9,10,13,14,15,16,17,21,22]). The papers [6,7,12] examine decomposability over more general rings, in the wake of the following result of Bilu and Tichy [3]: for f, g ∈ R[x], where R is the ring of S-integers of a number field, if the equation f (u) = g(v) has infinitely many solutions u, v ∈ R then f and g have decompositions of certain types.…”

Co-fibered products of algebraic curves

“…In positive characteristic, related questions on intersections of some specific function fields have been considered in [Ber73], [BM78], [Wat04], [BWZ07], [ZM08]. A sample result from [Ber73] is: If k is perfect field of characteristic p then k(x pn + x pn−1 ) ∩ k(x n ) = k if and only if gcd(p, n) = 1.…”