关于正特征函数域上多项式的定义
On definitions of polynomials over function fields of positive characteristi
摘要：
本文研究正特征函数域上多项式的定义问题。在其他结果中，我们证明了以下断言是正确的。1.设$\G_p$是$p$元素域的代数扩张，并假定$\G_p$不是代数闭的。设$t$是$\G_p$上的超越，$K$是$\G_p(t)$的有限扩张.在这种情况下，$\G_p[t]$具有形式为$\forall \exists \ldots \exists P$的$K$上的定义（带有参数），其中只有一个变量在全称量词的范围内，并且$P$是$K$上的多项式。2.对于任何$q$，对于所有$p \not=q$和如上所述的所有函数字段$K$，其中$\G_p$具有次数$q$的扩展和单位的本原$q$次根，在$p$和$K$定义中存在一致性（带参数）$\G_p[t]$，形式为$\exists \ldots \exists \forall \forall \exists \ldots \exists P$，在全称量词范围内只有两个变量，$P$是一个有限的析取和合取集合，$\Z/p$上的多项式方程。此外，对于任何固定大小$m$的$K$的素数的有限集合$\calS_K$，在$K$和$p$中存在$\calS_K$的环的统一定义-$\forall\forall\exists \ldots \exists P$形式的整数，其全称量词和$P$的范围如上所述。3.设$M$是任意常数域$H,$上的单变量$t$的正特征函数域，$\G_p$是$H$中有限域的代数闭包.假设$\G_p$不是代数闭的。在这种情况下，$\G_p[t]$在$M$上是一阶可定义的。
英文摘要：
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 $M$.
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