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Representation of powers by polynomials and the language of powers
Author(s) -
Garcia-Fritz Natalia
Publication year - 2013
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jds052
Subject(s) - mathematics , sums of powers , order (exchange) , representation (politics) , constant (computer programming) , field (mathematics) , power (physics) , expressive power , pure mathematics , discrete mathematics , combinatorics , computer science , physics , law , politics , political science , finance , quantum mechanics , theoretical computer science , economics , programming language
Given a field F and polynomials a and b in F [ t ], we prove that in general the sets { a λ+ b : λ∈ F } and {λ 2 + a λ+ b : λ∈ F } contain only finitely many powers and find bounds that are uniform from various points of view. We derive from this analysis various first‐order definability and undecidability results. For example, we prove that there is no algorithm to decide whether or not an arbitrary system of linear equations over the integers, together with conditions of the form ‘ x is a power’ and of the form ‘ x is non‐constant’ on some of the variables, has a solution in F [ t ].