Divisors of second-order sequences

Second order theta divisors on Pryms

The Schottky problem is the problem of finding necessary and sufficient conditions for a principally polarized abelian variety (ppav) to be a product of jacobians of smooth curves. Let (P,Ξ) be a ppav of dimension p ≥ 4 with Ξ a symmetric theta divisor on P . The elements of Γ = H(P, 2Ξ) are symmetric, hence their multiplicities at the origin are always even. Let Γ0 ⊂ Γ be the subvector space o...

Primitive Prime Divisors of First-Order Polynomial Recurrence Sequences

The question of which terms of a recurrence sequence fail to have primitive prime divisors has been significantly studied for several classes of linear recurrence sequences and for elliptic divisibility sequences. In this paper, we consider the question for sequences generated by the iteration of a polynomial. For two classes of polynomials f(x) ∈ Z[x] and initial values a1 ∈ Z, we show that th...

Reciprocal Sums of Second Order Recurrent Sequences

Let Z and IR (C) denote the ring of the integers and the field of real (complex) numbers, respectively. For a field F, we put i* = F\{0}. Fix A GC and B G C*, and let X(A, B) consist of all those second-order recurrent sequences {wn}neZ of complex numbers satisfying the recursion: n+i = „ ~ n-i (i.e., Bwn_Y = Awn wn+l) for n = 0, ± 1, ± 2,.... (1) For sequences in SE(A,B), the corresponding cha...

Intersections of Second - Order Linear Recursive Sequences

Positivity of second order linear recurrent sequences

We give an elementary proof for the Positivity Problem for second order recurrent sequences: it is decidable whether or not a recurrent sequence defined by un = aun−1 + bun−2 has only nonnegative terms.