Letter to the editor

Translated from Matematicheskie Zametki, Vol. 49, No. 6, p. 148, June, 1991.     

The Agnihotri—Woodward—Belkale polytope and Klyachko cones

The Agnihotri—Woodward—Belkale polytope Δ (resp., the Klyachko cone ) is the set of solutions of the multiplicative (resp., additive) Horn problem, i.e., the set of triples of spectra of special unitary (resp. traceless Hermitian) n × n matrices satisfying AB = C (resp. A + B = C). The set is the tangent cone of Δ at the origin. The group G = ? _n ⊕ ? _n acts naturally on Δ. In this note, we report on a computer calculation showing that Δ coincides with the intersection of , g ∈ G, for n ≤ 14 but does not coincide with it for n = 15. Our motivation was an attempt to understand how to solve the multiplicative Horn problem in practice for given conjugacy classes in SU(n).     

Asymptotic Growth of the Number of Classes of Real Plane Algebraic Curves as the Degree Grows

The work presents some results on the asymptotics of the number of real plane algebraic curves as the degree grows. In particular, we obtain the asymptotics of the number of curves considered up to the isotopy and rigid isotopy, as well as the number of isotopic classes of maximal curves realizable by T-curves. Some results are generalized to hypersurfaces in nonsingular algebraic varieties of arbitrary dimension. Bibliography: 18 titles.     

On commutator subgroups of Artin groups

Doklady Mathematics -     

On Alternating Quasipositive Links

Doklady Mathematics - An effectively verifiable condition for quasipositivity of links is given. In particular, it is proven that if a quasipositive link can be represented by an alternating...     

Multivariate Signatures of Iterated Torus Links

Functional Analysis and Its Applications - We compute the multivariate signatures of any Seifert link (that is, a union of some fibers in a Seifert homology sphere), in particular, of the union of...