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Department of Mathematical Sciences

# Seminar Archives

## Gandalf (Pure Maths Student Seminar): Integer valued polynomials

Presented by Anna Szumowicz, Durham

27 April 2016 16:00 in CM221

The polynomial ${X \choose m}$, where $m$ is a natural number is an example of a polynomial which takes integer values on $\mathbb{Z}$ even though its coefficients are not integer. A polynomial $f\in \mathbb{Q}[x]$ with the property $f(\matbb{Z}\subset \mathbb{Z})$ is called integer-valued. If $f$ is of degree at most $n$, then it is enough to check that $f({0,...n})$ takes values in $\mathbb{Z}$ to know that $f$ is integer-valued. A finite set $A$ is called $n$-universal if $f(A)\subset \mathbb{Z}$ implies that $f$ is integer-valued for every $f$ of degree at most $n$. I will talk about $n$-universal sets when $\mathbb{Q}$ is replaced by a number field.