Celebratio Mathematica

L. van den Dries and A. Günaydın: “The fields of real and complex numbers with a small multiplicative group,” Proc. London Math. Soc. (3) 93 : 1 (July 2006), pp. 43–81. MR 2235481 Zbl 1101.03028 article

L. van den Dries and A. Günaydın: “Mann pairs,” Trans. Am. Math. Soc. 362 : 5 (2010), pp. 2393–2414. MR 2584604 Zbl 1192.03011 article

Mann proved in the 1960s that for any \( n \geq 1 \) there is a finite set \( E \) of \( n \)-tuples \( (\eta_1,\dots \), \( \eta_n) \) of complex roots of unity with the following property: if \( a_1,\dots \), \( a_n \) are any rational numbers and \( \zeta_1,\dots \), \( \zeta_n \) are any complex roots of unity such that \[ \sum_{i=1}^n a_i\zeta_i = 1 \quad\text{and}\quad \sum_{i\in I} a_i \zeta_i\neq 0 \] for all nonempty \( I\subseteq \{1,\dots,n\} \), then \[ (\zeta_1,\dots,\zeta_n)\in E .\] Taking an arbitrary field \( \mathbf{k} \) instead of \( \mathbb{Q} \) and any multiplicative group in an extension field of \( \mathbf{k} \) instead of the group of roots of unity, this property defines what we call a Mann pair \( (\mathbf{k},\Gamma) \). We show that Mann pairs are robust in certain ways, construct various kinds of Mann pairs, and characterize them model-theoretically.

L. van den Dries and A. Günaydın: “Erratum to ‘Mann pairs’,” Trans. Am. Math. Soc. 363 : 9 (2011), pp. 5057. Erratum to an article published in Trans. Am. Math. Soc. 362:5 (2010). MR 2806701 Zbl 1227.03049 article

L. van den Dries and A. Günaydın: “Definable sets in Mann pairs,” Comm. Algebra 39 : 8 (2011), pp. 2752–2763. MR 2834128 Zbl 1243.03054 article

Consider structures \( (\Omega,\mathbf{k},\Gamma) \) where \( \Omega \) is an algebraically closed field of characteristic zero, \( \mathbf{k} \) is a subfield, and \( \Gamma \) is a subgroup of the multiplicative group of \( \Omega \). Certain pairs \( (\mathbf{k},\Gamma) \) have been singled out as Mann pairs in [van den Dries and Günaydın 2010]. We give new examples of such Mann pairs, we axiomatize for each Mann pair \( (\mathbf{k},\Gamma) \) the first-order theory of \( (\Omega,\mathbf{k},\Gamma) \) in a cleaner way than in [van den Dries and Günaydın 2010], and, as the main result of the article, we characterize the subsets of \( \Omega^n \) that are definable in \( (\Omega,\mathbf{k},\Gamma) \).

year	title	people

			clear

Lou van den Dries

Complete Bibliography

Works connected to Ayhan Günaydın

Filter the Bibliography List