Rings of integer-valued rational functions

Rings of functions with integer derivatives at x=1

Generalized Rings of Integer-valued Polynomials

The classical ring of integer-valued polynomials Int(Z) consists of the polynomials in Q[X] that map Z into Z. We consider a generalization of integervalued polynomials where elements of Q[X] act on sets such as rings of algebraic integers or the ring of n× n matrices with entries in Z. The collection of polynomials thus produced is a subring of Int(Z), and the principal question we consider is...

What are Rings of Integer-Valued Polynomials?

Every integer is either even or odd, so we know that the polynomial f(x) = x(x− 1) 2 is integervalued on the integers, even though its coefficients are not in Z. Similarly, since every binomial coefficient ( k n ) is an integer, the polynomial ( x n ) = x(x− 1)...(x− n+ 1) n! must also be integervalued. These polynomials were used for polynomial interpolation as far back as the 17 century. Inte...

Integer-valued continuous functions

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Integer-valued definable functions

We present a dichotomy, in terms of growth at infinity, of analytic functions definable in the real exponential field which take integer values at natural number inputs. Using a result concerning the density of rational points on curves definable in this structure, we show that if a function f : [0,∞) → R is such that f(N) ⊆ Z, then either sup|x̄|≤r f(x̄) grows faster than exp(r), for some δ > 0,...