Submission date: 9 February 2015

Abstract:

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 ... \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 \neq 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 ... \exists \forall \forall \exists ... \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 S_K of primes of K of fixed size m, there is a uniform in K and p definition of the ring of S_K-integers of the form \forall\forall\exists ... \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.

Mathematics Subject Classification: 11U09

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Full text arXiv 1502.02714: pdf, ps.