2015/02/09 by Alexandra Shlapentokh, Shlapentokh, Alexandra · 1 citation
Mathematics · #11U09 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #FOS: Mathematics #Logic (math.LO) #Meromorphic and Entire Functions #Number Theory (math.NT) #math.LO #math.NT #msc:11U09
paper · pdf · doi:10.48550/arxiv.1502.02714
arxiv created 2015/02/09 · openalex publication_date 2015/02/09 · arxiv updated 2015/02/11 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28
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 \Gp be an algebraic extension of a field of p elements and assume \Gp is not algebraically closed. Let t be transcendental over \Gp, and let K be a finite extension of \Gp(t). In this case \Gp[t] has a definition (with parameters) over K of the form ∀ ∃ … ∃ 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 \Gp 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 \Gp[t], of the form ∃ … ∃ ∀ ∀ ∃ … ∃ 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 \calSK of primes of K of fixed size m, there is a uniform in K and p definition of the ring of \calSK-integers of the form ∀∀∃ … ∃ 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 \Gp be the algebraic closure of a finite field in H. Assume \Gp is not algebraically closed. In this case \Gp[t] is first-order definable over M.