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We present lower bounds for the orbit length of reduction modulo primes of parametric polynomial dynamical systems defined over the integers, under a suitable hypothesis on its set of preperiodic points over C. Applying recent results of Baker and DeMarco (2011) and of Ghioca, Krieger, Nguyen and Ye (2017), we obtain explicit families of parametric polynomials and initial points such that the reductions modulo primes have long orbits, for all but a finite number of values of the parameters. This generalizes a previous lower bound due to Chang (2015). As a by-product, we also slightly improve a result of Silverman (2008) and recover a result of Akbary and Ghioca (2009) as special extreme cases of our estimates. [ABSTRACT FROM AUTHOR]

Let f(x) = ax² + bx + c ∈ Z[x] be a quadratic polynomial with a ≢ 0 mod p. Take z ∈ Fp and let Oz = {fi(z)}i∈Z+ be the orbit of z under f, where fi(z) = f(fi-1(z)) and f0(z) = z. For M < |Oz|, we study the diameter of the partial orbit OM = {z, f(z), f2(z),..., fM-1(z)} and prove that there exists c1 > 0 such that diam OM ≳ min {...}. For a complete orbit C, we prove that diam C ≳ min{p5c1, eT/4}, where T is the period of the orbit. [ABSTRACT FROM AUTHOR]

A vector bundle has the {\em Bloch-Gieseker property} if all its Chern classes are numerically positive. In this paper we show that the non-ample bundle $\Omega ^{p}_{\mathbb{P}_{n}}(p+1)$ has the Bloch-Gieseker property, except for two cases, in which the top Chern classes are trivial and the other Chern classes are positive. Our method is to reduce the problem to showing, e.g. the positivity of the coefficient of $t^{k}$ in the rational function $\frac{(1+t)^{\binom n p} (1+3t)^{\binom {n}{p-2}} \cdots (1+(p-1)t)^{\binom n2} (1+(p+1)t)}{(1+2t)^{\binom {n}{p-1}} (1+4t)^{\binom {n}{p-3}} \cdots (1+pt)^{\binom {n}{1}}}$ (for $p$ even). [ABSTRACT FROM AUTHOR]