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The article discusses the algorithm for the counting of the irreducible elements of a complex algebraic variety defined by a fixed number of polynomials encoded as straight-line programs (slps) in mathematics. The algorithm runs in a polynomial time in the Blum-Shub-Smale (BSS) model and in randomized parallel polylogarithmic time in the Turing model. It presents the general complexity theoretic framework needed for algorithms in algebraic geometry.

This paper classifies a class of holomorphic D-branes, closely related to framed torsion-free sheaves, on threefolds fibered in resolved ADE surfaces over a general curve C, in terms of representations with relations of a twisted Kronheimer–Nakajima-type quiver in the category Coh( C) of coherent sheaves on C. For the local Calabi–Yau case $$C\cong{\bf A}^1$$ and special choice of framing, one recovers the N = 1 ADE quiver studied by Cachazo–Katz–Vafa. [ABSTRACT FROM AUTHOR]

Consider a projective algebraic variety W that is an irreducible component of the set of all common zeros of a family of homogeneous polynomials of degrees less than d in n + 1 variables in zero characteristic. Consider a dominant rational morphism from W to W′ given by homogeneous polynomials of degree d′. We suggest algorithms for constructing objects in general position related to this morphism. These algorithms are deterministic and polynomial in (dd′)n and the size of the input. Bibliography: 13 titles. [ABSTRACT FROM AUTHOR]

Recently, V. Chernousov, S. Gille and A. Merkurjev have obtained a decomposition of the motive of an isotropic smooth projective homogeneous variety analogous to the Bruhat decomposition. Using the method of A. Bialynicki-Birula and a corollary, which is essentially due to S. del Baño, I generalize this decomposition to the case of a (possibly anisotropic) smooth projective variety homogeneous under the action of an isotropic reductive group. This answers a question of N. Karpenko. [ABSTRACT FROM AUTHOR]

The paper presents examples of complete singular toric varieties associated to the Niemeier lattices. The singularities and automorphisms of these varieties are seen to be closely related to the Golay codes. [ABSTRACT FROM AUTHOR]

Let X be a smooth contractible three-dimensional affine algebraic variety with a free algebraic C+-action on it such that S=X//C+ is smooth. We prove that X is isomorphic to S×C and the action is induced by a translation on the second factor. As a consequence we show that any free algebraic C+-action on C3 is a translation in a suitable coordinate system. [ABSTRACT FROM AUTHOR]

A formula of type indicated in the title is presented and discussed. Bibliography: 7 titles. [ABSTRACT FROM AUTHOR]