Finiteness properties of differential polynomials

Finiteness for Degenerate Polynomials

Let MPd denote the space of polynomials f : C → C of degree d ≥ 2, modulo conjugation by Aut(C). Using properties of polynomial trees (as introduced in [DM]), we show that if fn is a divergent sequence of polynomials in MPd, then any subsequential limit of the measures of maximal entropy m(fn) will have finite support. With similar techniques, we observe that the iteration maps {MPd 99K MPdn : ...

Finiteness results for regular ternary quadratic polynomials

In 1924, Helmut Hasse established a local-to-global principle for representations of rational quadratic forms. Unfortunately, an analogous local-to-global principle does not hold for representations over the integers. A quadratic polynomial is called regular if such a principle exists; that is, if it represents all the integers which are represented locally by the polynomial itself over Zp for ...

Finiteness Properties of Database Queries

We investigate the problem of checking whether the number of derivation trees of a Datalog program with duplicate semantics is nite or not. We show that given a safe stratiied query and an edb, it is possible to check, in polynomial time, whether the query has a nite number of derivation trees. However, it is undecidable to check whether a safe stratiied query has a nite number of derivation tr...

Fractional differential equations with Legendre polynomials

A Finiteness Property for Preperiodic Points of Chebyshev Polynomials

Let K be a number field with algebraic closure K, let S be a finite set of places of K containing the archimedean places, and let φ be a Chebyshev polynomial. We prove that if α ∈ K is not preperiodic, then there are only finitely many preperiodic points β ∈ K which are S-integral with respect to α.