Polynomial-time computation of the degree of a dominant morphism in zero characteristic. II

Consider a projective algebraic variety W which 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 over a field of 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. They generalize some algorithms from the first part of the paper to the case dim W > dim W′. These algorithms are deterministic and polynomial in (dd′)n and the size of the input. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 181–224.     

Polynomial-time computation of the degree of a dominant morphism in zero characteristic. III

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. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 344, 2007, pp. 203–239.     

Estimating the first zero of a characteristic function

For a characteristic function (Fourier transform of a probability distribution), the first zero encodes important information. We present a general lower bound estimation of the first zero in terms of a moment of any order. The result proves the complementary nature between the first zero and moments, and has interesting implications for quantum mechanical uncertainty relations. To cite this article: S. Luo, Z. Zhang, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Rings containing a field of characteristic zero

Let K be a field of characteristic zero, and let R be a ring containing K. Then either \(R^\times = K^\times \) or \(K^\times \) is a subgroup of infinite index in \(R^\times \).     

Simple Hironaka resolution in characteristic zero

Building upon work of Villamayor and Bierstone-Milman we give a proof of the canonical Hironaka principalization and desingularization. The idea of ``homogenized ideals" introduced in the paper gives a priori the canonicity of the algorithm and radically simplifies the proof.

     

Desingularization of quasi-excellent schemes in characteristic zero

Grothendieck proved in EGA IV that if any integral scheme of finite type over a locally noetherian scheme X admits a desingularization, then X is quasi-excellent, and conjectured that the converse is probably true. We prove this conjecture for noetherian schemes of characteristic zero. Namely, starting with the resolution of singularities for algebraic varieties of characteristic zero, we prove the resolution of singularities for noetherian quasi-excellent Q-schemes.     

Accurate computation of the smallest eigenvalue of a diagonally dominant -matrix

If each off-diagonal entry and the sum of each row of a diagonally dominant -matrix are known to certain relative accuracy, then its smallest eigenvalue and the entries of its inverse are known to the same order relative accuracy independent of any condition numbers. In this paper, we devise algorithms that compute these quantities with relative errors in the magnitude of the machine precision. Rounding error analysis and numerical examples are presented to demonstrate the numerical behaviour of the algorithms.

     

On a singular point with one zero characteristic root

A sufficiently smooth real autonomous system of differential equations with an isolated singular point on the plane is considered. A new method for investigating the behavior of trajectories of the system in the neighborhood of this point is suggested.     

Characteristic subgroups of the Jonquere group over a field of characteristic zero

The lattice of characteristic subgroups of the Jonquere group over a field of characteristic zero is described.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 111–113, January, 1995.     

Normal structure of the jonquere group over a field of characteristic zero

The lattice of normal divisors of the Jonquere group over a field of characteristic zero is described. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 692–698, June, 1994.     

The polynomial identities of matrices in characteristic zero

The multilinear identities of the k×k matrices F₂ are studied in terms of the cocharacters and codimension sequences. For F₂ the estimates are fairly good, while for k≥3 only partial results are obtained.     

The edge-face coloring of graphs embedded in a surface of characteristic zero

Let G be a graph embedded in a surface of characteristic zero with maximum degree Δ. The edge-face chromatic number χ_ef(G) of G is the least number of colors such that any two adjacent edges, adjacent faces, incident edge and face have different colors. In this paper, we prove that χ_ef(G)≤Δ+1 if Δ≥13, χ_ef(G)≤Δ+2 if Δ≥12, χ_ef(G)≤Δ+3 if Δ≥4, and χ_ef(G)≤7 if Δ≤3.     

Irreducible modules over a Jordan plane over a field of characteristic zero

Irreducible modules over a Jordan plane over a field of characteristic zero are considered. All finite-dimensional modules are described. Examples of infinite-dimensional irreducible modules are constructed.