### Working paper

## Weighted PBW degenerations and tropical flag varieties

It has been shown recently that the normalized median Genocchi numbers are equal to the Euler characteristics of the degenerate flag varieties. The q-analogues of the Genocchi numbers can be naturally defined as the Poincare polynomials of the degenerate flag varieties. We prove that the generating function of the Poincare polynomials can be written as a simple continued fraction. As an application we prove that the Poincare polynomials coincide with the q-version of the normalized median Genocchi numbers introduced by Han and Zeng.

Toric geometry exhibited a profound relation between algebra and topology on one side and combinatorics and convex geometry on the other side. In the last decades, the interplay between algebraic and convex geometry has been explored and used successfully in a much more general setting: first, for varieties with an algebraic group action (such as spherical varieties) and recently for all algebraic varieties (construction of Newton-Okounkov bodies). The main goal of the conference is to survey recent developments in these directions. Main topics of the conference are: Theory of Newton polytopes and Newton-Okounkov bodies; Toric geometry, geometry of spherical varieties, Schubert calculus, geometry of moduli spaces; Tropical geometry and convex geometry; Real algebraic geometry and fewnomial theory; Polynomial vector fields and the Hilbert 16th problem.

This volume contains the proceedings of the International Workshop on Tropical and Idempotent Mathematics, held at the Independent University of Moscow, Russia, from August 26-31, 2012. The main purpose of the conference was to bring together and unite researchers and specialists in various areas of tropical and idempotent mathematics and applications. This volume contains articles on algebraic foundations of tropical mathematics as well as articles on applications of tropical mathematics in various fields as diverse as economics, electroenergetic networks, chemical reactions, representation theory, and foundations of classical thermodynamics. This volume is intended for graduate students and researchers interested in tropical and idempotent mathematics or in their applications in other areas of mathematics and in technical sciences

Suppose that there exists a hypersurface with the Newton polytope Δ, which passes through a given set of subvarieties. Using tropical geometry, we associate a subset of Δ to each of these subvarieties. We prove that a weighted sum of the volumes of these subsets estimates the volume of Δ from below. As a particular application of our method we consider a planar algebraic curve *C* which passes through generic points p1,…,pn with prescribed multiplicities m1,…,mn. Suppose that the minimal lattice width ω(Δ) of the Newton polygon Δ of the curve *C* is at least max(mi). Using tropical floor diagrams (a certain degeneration of p1,…,pn on a horizontal line) we prove that

area(Δ)≥12n∑i=1m2i−S, where S=12max(n∑i=1s2i|si≤mi,n∑i=1si≤ω(Δ)).

In the case m1=m2=⋯=m≤ω(Δ) this estimate becomes area(Δ)≥12(n−ω(Δ)m)m2. That rewrites as d≥(√n−12−12√n)m for the curves of degree *d*. We consider an arbitrary toric surface (i.e. arbitrary Δ) and our ground field is an infinite field of any characteristic, or a finite field large enough. The latter constraint arises because it is not *a priori* clear what is *a collection of generic points* in the case of a small finite field. We construct such collections for fields big enough, and that may be also interesting for the coding theory.

Tropical algebra is an emerging field with a number of applications in various areas of mathematics. In many of these applications appeal to tropical polynomials allows to study properties of mathematical objects such as algebraic varieties and algebraic curves from the computational point of view. This makes it important to study both mathematical and computational aspects of tropical polynomials. In this paper we prove tropical Nullstellensatz and moreover we show effective formulation of this theorem. Nullstellensatz is a next natural step in building algebraic theory of tropical polynomials and effective version is relevant for computational aspects of this field.

We study geometric and combinatorial properties of the degenerate flag varieties of type A. These varieties are acted upon by the automorphism group of a certain representation of a type A quiver, containing a maximal torus T . Using the group action, we describe the moment graphs, encoding the zero- and one-dimensional T -orbits. We also study the smooth and singular loci of the degenerate flag varieties. We show that the Euler characteristic of the smooth locus is equal to the large Schröder number and the Poincaré polynomial is given by a natural statistics counting the number of diagonal steps in a Schröder path. As an application we obtain a new combinatorial description of the large and small Schröder numbers and their q-analogues.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.