arXiv:2009.03387v1 [math.RA] 7 Sep 2020 FIRST-ORDER CHARACTERIZATION OF NONCOMMUTATIVE BIRATIONAL EQUIVALENCE HUGO LUIZ MARIANO, JO ˜ AO SCHWARZ Abstract. Let Σ be a root system with Weyl group W . Let k be an al- gebraically closed field of zero characteristic, and consider the corresponding semisimple Lie algebra g k,Σ . Then there is a first-order sentence φ Σ in the language L = (1, 0, +, ∗) of rings sucht that, for any algebraically closed field k of char = 0, the validity of the Gelfand-Kirillov Conjecture for g k,Σ is equiv- alent to ACF 0 ⊢ φ Σ . By the same method, we can show that the validity of Noncommutative Noether’s Problem for An(k) W , k any algebraically closed field of char = 0 is equivalent to ACF 0 ⊢ φ W , φ W a formula in the same language. As consequences, we obtain results on the modular Gelfand-Kirillov Conjecture and we show that, for F algebraically closed with characteristic p >> 0, An(F) W is a case of positive solution of modular Noncommutative Noether’s Problem. 1. Introduction Connections between Algebra and Logic are well known (cf. [10], [34]). In the specific topic of Algebraic Geometry, this line of inquiry began with the work of Alfred Tarski on the decidability via quantifier elimination in the theory of alge- braically closed fields, and have achieved a remarkable development through the years using more sophisticated methods of Model Theory in Algebraic Geometry, such as the work of Ax, Kochen and Ershov on Artin’s Conjecture, or the celebrated proof of Mordell-Lang Conjecture by Hrushovski (cf. [27], [8], [24], [35]). Let’s fix some conventions. All our rings and fields will be algebras over a basis field k. One of the main problems of algebraic geometry is the birational classification of varieties ([25]). A particular important example for us is Noether’s Problem ([39]), which asks: Let G be a finite group acting linearly on k(x 1 ,...,x n ) G . When is k(x 1 ,...,x n ) G isomorphic to k(x 1 ,...,x n )? In algebraic-geometric terms: when is the va- riety k n /G rational? This is a very important area of research, given its connection to the inverse Galois problem (cf. [31]), PI-algebras (cf. [14]), study of moduli spaces (cf. [11]). In the 1966 the study of birational geometry of noncommutative objects began. In his adress at the 1966 ICM in Moscow, A. A. Kirillov proposed to classify, up to birational equivalence, the enveloping algebras U (g) of finite dimensional algebraic Lie algebras g when k is algebraically closed of zero characteristic. This means to MSC 2020 Primary: 03C60; Secundary: 16S85, 16W22, 17B35 Keywords: First-order characterization, noncommutative birational equivalence, Gelfand- Kirillov Conjecture 1