Notes on the decidability of addition and the Frobenius map for polynomials and rational functions

• Published in: Reports on mathematical logic Vol. 57; no. 57; pp. 53 - 60
• Main Authors: Chompitaki, Dimitra, Kamarianakis, Manos, Pheidas, Thanases
• Format: Journal Article
• Language: English
• Published: Kraków Wydawnictwo Uniwersytetu Jagiellońskiego 01.01.2022; Jagiellonian University Press; Jagiellonian University-Jagiellonian University Press
• Subjects: Algebra; Algorithms; Completeness; Fields (mathematics); Logic; Mathematical analysis; Philosophy of Science; Polynomials; Prime numbers; Questions; Rational functions; Rings (mathematics); Variables; polynomial rings; model completeness; Frobenius map; decidability; rational functions
• ISSN: 0137-2904; 2084-2589
• DOI: 10.4467/20842589RM.22.004.16661

Let pbe a prime number, Fp a finite field with pelements, Fan algebraic extension of Fp and z a variable. We consider the structure of addition and the Frobenius map (i.e., x →xp) in the polynomial rings F[z] and in fields F(z) of rational functions. We prove that any question about F[z] in the structure of addition and Frobenius map may be effectively reduced to questions about the similar structure of the field F. Furthermore, we provide an example which shows that a fact which is true for addition and the Frobenius map in the polynomial rings F[z] fails to be true in F(z). As a consequence, certain methods used to prove model completeness for polynomials do not suffice to prove model completeness for similar structures for fields of rational functions F(z), a problem that remains open even for F= Fp.