  We consider the problem of defining polynomials over function fields of
positive characteristic. Among other results, we show that the following
assertions are true.
  1. Let $\G_p$ be an algebraic extension of a field of $p$ elements and assume
$\G_p$ is not algebraically closed. Let $t$ be transcendental over $\G_p$, and
let $K$ be a finite extension of $\G_p(t)$. In this case $\G_p[t]$ has a
definition (with parameters) over $K$ of the form $\forall \exists \ldots
\exists P$ with only one variable in the range of the universal quantifier and
$P$ being a polynomial over $K$.
  2. For any $q$, for all $p \not=q$ and all function fields $K$ as above with
$\G_p$ having an extension of degree $q$ and a primitive $q$-th root of unity,
there is a uniform in $p$ and $K$ definition (with parameters) of $\G_p[t]$, of
the form $\exists \ldots \exists \forall \forall \exists \ldots \exists P$ with
only two variables in the range of universal quantifiers and $P$ being a finite
collection of disjunction and conjunction of polynomial equations over $\Z/p$.
Further, for any finite collection $\calS_K$ of primes of $K$ of fixed size
$m$, there is a uniform in $K$ and $p$ definition of the ring of
$\calS_K$-integers of the form $\forall\forall\exists \ldots \exists P$ with
the range of universal quantifiers and $P$ as above.
  3. Let $M$ be a function field of positive characteristic in one variable $t$
over an arbitrary constant field $H,$ and let $\G_p$ be the algebraic closure
of a finite field in $H$. Assume $\G_p$ is not algebraically closed. In this
case $\G_p[t]$ is first-order definable over $M$.
