Implicit function theorem over free groups
By Olga Kharlampovich, Alexei G. Myasnikov. 2003
78 citations recorded by the source corpus.
Abstract We introduce the notion of a regular quadratic equation and a regular NTQ system over a free group. We prove the results that can be described as implicit function theorems for algebraic varieties corresponding to regular quadratic and NTQ systems. We will also show that the implicit function theorem is true only for these varieties. In algebraic geometry such results would be described as lifting solutions of equations into generic points. From the model theoretic view-point we claim the existence of simple Skolem functions for particular ∀∃-formulas over free groups. Proving these theorems we describe in details a new version of the Makanin–Razborov process for solving equations in free groups. We also prove a weak version of the implicit function theorem for NTQ systems which is one of the key results in the solution of the Tarski's problems about the elementary theory of a free group. We call it the parametrization theorem.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/01bace6d9b5bfeb6e50d8d9d4b0c3ec4539f4bd1
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In Geometry and Topology
- Henry Wilton
- Vincent Guirardel
- Alexei G. Myasnikov
- Olga Kharlampovich
- Franccois Dahmani
- Denis V. Osin
- D. Groves
- Alessandro Sisto
In Mathematical Physics
- Henry Wilton
- Vincent Guirardel
- Alexei G. Myasnikov
- Olga Kharlampovich
- Franccois Dahmani
- Denis V. Osin
- Gilbert Levitt
- Alessandro Sisto
Citations in the corpus
- Effective JSJ Decompositions (2004, methodology): “In [36] and [40], Section 5, we described a general process of transformations of a generalized equation and construction of fundamental sequences of solutions of equations.”
- Hall’s Theorem for Limit Groups (2006, background): “Miasnikov also announced a solution to Tarski’s Problem in [11, 12, 14, 13, 15].”
- Elementary theory of free non-abelian groups (2006, methodology): “Recall, that in the papers [13,14] instead of inserting dummy variables from T in the equations Si it was simply assumed that there is an F -homomorphism πn :Gn → F ∗F(T ) and it was allowed to use arbitrary specializations η :F ∗ F(T ) → F .”
- Foliations for solving equations in groups: free, virtually free, and hyperbolic groups (2009, background): “An interesting and well-written account on Makanin’s algorithm for equations in free monoids (a simplified version of the case of free groups, which does not imply a solution for free groups) was given in [15], and another one on Makanin and Razborov’s algorithm for free groups was given in [26].”