Exponential prefixed polynomial equations

A prefixed polynomial equation is an equation of the form \(P(t_1,\ldots,t_n) = 0\), where \(P\) is a polynomial whose variables \(t_1,\ldots,t_n\) range over the natural numbers, preceded by quantifiers over some, or all, of its variables. Here, we consider exponential prefixed polynomial equations...

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Bibliographic Details
Published in:arXiv.org 2013-03
Main Author: Nayebi, Aran
Format: Article
Language:English
Subjects:
Arithmetic
Combinatorial analysis
Equivalence
Mathematical analysis
Number theory
Polynomials
Theorems
Online Access:Get full text
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Summary:A prefixed polynomial equation is an equation of the form \(P(t_1,\ldots,t_n) = 0\), where \(P\) is a polynomial whose variables \(t_1,\ldots,t_n\) range over the natural numbers, preceded by quantifiers over some, or all, of its variables. Here, we consider exponential prefixed polynomial equations (EPPEs), where variables can also occur as exponents. We obtain a relatively concise EPPE equivalent to the combinatorial principle of the Paris-Harrington theorem for pairs (which is independent of primitive recursive arithmetic), as well as an EPPE equivalent to Goodstein's theorem (which is independent of Peano arithmetic). Some new devices are used in addition to known methods for the elimination of bounded universal quantifiers for Diophantine predicates.
ISSN:2331-8422