Andrei Bengus-Lasnier,
IMI-BAS

Abstract:

Non-abelian analogs of tropicalizations have been considered for families X(t) for t>0 of curves and surfaces inside SL₂(C). Roughly speaking, we look at asymptotics of germs of analytic (or convergent Puiseux or Hahn) functions at infinity A(t) ∈ X(t) for t>0. Such functions form a valued field (K, ν), and X can be seen as an algebraic variety X = V(I) for an ideal I ⊂ K[x₁, …, xₙ].

In this talk I will introduce valuation-theoretic tools, such as the graded algebra grν(R) of a general valuation ν on a commutative ring R. This will serve as a tool to algebraically encode the leading terms of our analytic solutions. The graded algebra construction presents certain advantages, such as functoriality, which is essential for proving the following lifting theorem.

For any fixed value γ, the leading terms of order γ will satisfy INγ(I), the ideal generated by the initial forms of the equations defining X. This will be an ideal in grνγ(K[x₁, …, xₙ]) for the monomial valuation νγ defined by the tuple (γ, …, γ). In turn, any point in V(INγ(I)) can be lifted to a solution in X. Thus, by studying the algebraic properties of these ideals INγ(I), one can deduce geometric properties of these SL₂ tropicalizations.

This work is in collaboration with Mikhail Shkolnikov.