[Paper Review] On definitions of polynomials over function fields of positive characteristi

Alexandra Shlapentokh|arXiv (Cornell University)|Feb 9, 2015

Algebraic Geometry and Number Theory14 references3 citations

TL;DR

This paper establishes first-order definability results for polynomial rings and rings of S-integers over function fields in positive characteristic. It proves that 𝔾ₚ[t] is definable over finite extensions K of 𝔾ₚ(t) using ∀∃-formulas with one universal variable, and provides uniform ∀∀∃…∃-definitions for 𝔾ₚ[t] and S-integers across varying p and K under certain conditions, advancing the model theory of function fields in positive characteristic.

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 $\forall \exists \ldots \exists P$ with only one variable in the range of the universal quantifier and $P$ being a polynomial over $K$. 2. For any $q$, for all $p ot=q$ 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 $M$.

Motivation & Objective

• To investigate first-order definability of polynomial rings and S-integers over function fields in positive characteristic.
• To extend results analogous to Hilbert's Tenth Problem and J. Robinson's work on number fields to function fields of positive characteristic.
• To reduce the number of universal quantifiers in definability constructions, inspired by progress in number field settings.
• To establish uniform definability results across parameters p and K under specific algebraic conditions.
• To analyze the quantifier complexity of Rumely's formula for integral closures in function fields.

Proposed method

• Uses algebraic geometry and valuation theory over function fields of positive characteristic.
• Applies norm equations and p-power Frobenius constructions to define subrings via polynomial conditions.
• Employs the structure of finite extensions of 𝔾ₚ(t), where 𝔾ₚ is a non-closed algebraic extension of 𝔽_p.
• Constructs definitions using systems of polynomial equations over 𝔽_p, with quantifier alternation patterns.
• Analyzes Rumely's formula for integral closure and estimates the number of universal quantifiers in prenex normal form.
• Utilizes the existence of primitive q-th roots of unity and extensions of degree q to construct uniform definitions.

Experimental results

Research questions

• RQ1 Can the polynomial ring 𝔾ₚ[t] be defined over finite extensions K of 𝔾ₚ(t) in positive characteristic?
• RQ2 Is there a uniform ∀∀∃…∃-definition of 𝔾ₚ[t] across varying p and K when certain field-theoretic conditions (e.g., existence of q-th roots of unity) are met?
• RQ3 Can rings of S-integers be uniformly defined over function fields of positive characteristic with bounded S?
• RQ4 What is the minimal number of universal quantifiers required to define integral closures in function fields using Rumely’s approach?
• RQ5 Does the existence of infinite p-adically discrete Diophantine sets in positive characteristic enable simpler definability compared to number fields?

Key findings

• For any finite extension K of 𝔾ₚ(t), where 𝔾ₚ is a non-algebraically closed algebraic extension of 𝔽_p, the ring 𝔾ₚ[t] has a ∀∃-definition over K with only one variable in the universal quantifier range.
• When q ≠ p and 𝔾ₚ admits a degree-q extension with a primitive q-th root of unity, there exists a uniform ∀∀∃…∃-definition of 𝔾ₚ[t] over K, with only two universal variables.
• For any fixed size m, the ring of S_K-integers has a uniform ∀∀∃…∃-definition over K and p, with two universal variables and polynomial equations over 𝔽_p.
• The ring of integers O_x in a global function field is definable via a formula involving norm equations and valuation conditions, with quantifier complexity estimated at at least 16 universal variables in Rumely’s construction.
• The construction of definable subrings relies on the existence of infinite p-adically discrete Diophantine sets, which are possible in positive characteristic but not in number fields under Mazur’s conjecture.
• The paper shows that 𝔾ₚ[t] is first-order definable over any function field M in one variable over a constant field H, provided 𝔾_p is the algebraic closure of a finite field in H and is not algebraically closed.