complexity of bezout s theorem iv probability of success extensions we estimate the probability that a given number of projective newton steps applied to a linear homotopy of a system of n homogeneous polynomial equations in n 1 complex variables of fixed degrees will find all the roots of the system we also extend the framework of our analysis to cover the classical implicit function theorem and revisit the condition number in this context further complexity theory is developed

orientational ordering in two dimensional systems with long range interaction a variational method for calculating gaussian fluctuations of two dimensional 2d moment orientations in lattice systems with arbitrary anisotropic interaction is proposed it is asymptotically exact for very low temperatures and gives a good extrapolation for a certain low temperature range general relationships are obtained that enable one to represent from a unified standpoint and in some cases to refine the known results on 1 the existence of short range order in a degenerate 2d system of moments with isotropic short range interaction 2 long range order stabilization by weak local potentials and dipole interactions 3 the existence of long range order in a 2d system with interaction of the type v r r if 2 4 for systems with anisotropic long range dipole interaction of moments on arbitrary 2d lattices except square ones for which a short range phase takes place the existence of long range order is proved lower limits for temperatures of transitions into phases with long range order are evaluated the low values of the transition temperatures in the dipole systems can be explained due to the small values of interchain interaction in comparison with interchain ones the applicability area for the method used is discussed

computational complexity on the geometry of polynomials and a theory of cost ii this paper deals with traditional algorithms newton s method and a higher order generalization due to euler these iterations schemes and their modifications have had a great success in solving nonlinear systems of equations we give some understanding of this phenomenon by giving estimates of efficiency the problem we focus on is that of finding a zero of a complex polynomial

on simple zeros of analytic functions of n variables we give a numerical criterion for a badly conditioned zero of a system of analytic equations to be part of a cluster of two zeros

on the probability distribution of condition numbers of complete intersection varieties and the average radius of convergence of newton s method in the underdetermined case in these pages we show upper bound estimates on the probability distribution of the condition numbers of smooth complete intersection algebraic varieties as a by product we also obtain lower bounds for the average value of the radius of newton s basin of attraction in the case of positive dimension affine complex algebraic varieties

complexity of bezout s theorem v polynomial time we show that there are algorithms which find an approximate zero of a system of polynomial equations and which function in polynomial time on the average the number of arithmetic operations is cn 4s where n is the input size and c a universal constant

how many eigenvalues of a random matrix are real let a be an n x n matrix whose elements are independent random variables with standard normal distributions as n oo the expected number of real eigenvalues is asymptotic to v 7r we obtain a closed form expres sion for the expected number of real eigenvalues for finite n and a formula for the density of a real eigenvalue for finite n asymptotically a real normalized eigenvalue al ii of such a random matrix is uniformly distributed on the in terval 1 1 analogous but strikingly different results are presented for the real generalized eigenvalues we report on numerical experiments confirming these results and suggesting that the assumption of normality is not important for the asymptotic results department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address edelmanfmath mit edu arts and sciences kapi olani community college 4303 diamond head road honolulu hawaii 98616 e mail address kostlanfkahuna math hawaii edu tj watson rsearch center 32 2 ibm yorktown heights new york 10598 0218 e mail address shubqwatson ibm com this content downloaded from 157 55 39 29 on tue 12 apr 2016 08 54 01 utc all use subject to http about jstor org terms

kloosterman sums for clifford algebras and a lower bound for the positive eigenvalues of the laplacian for congruence subgroups acting on hyperbolic spaces

on the representation of integers by the lorentzian quadratic form in this paper we establish an asymptotic formula for large radiusr for the number of representations of a nonzero integerkby the lorentzian quadratic formx21 x22 x2n x2n 1that are contained in the ball of radiusrcentered at the origin in euclidean n 1 space

solving a polynomial equation some history and recent progress the classical problem of solving an nth degree polynomial equation has substantially influenced the development of mathematics throughout the centuries and still has several important applications to the theory and practice of present day computing we briefly recall the history of the algorithmic approach to this problem and then review some successful solution algorithms we end by outlining some algorithms of 1995 that solve this problem at a surprisingly low computational cost

the circle problem in the hyperbolic plane abstract this paper deal with the analogue of the classical circle problem in the hyperbolic plane that is we count the number n s z of translates of a base point z by a fuchsian group which lie in a geodesic ball of radius s about z if s z is the theoretical best approximation to n s z which has e s vol as its leading term we set d s z n s z s z the best known upper bound for which is o e 2 s 3 we get omega results for d s z which in the co compact case are d s z e s 2 s where s as s we also show that the normalized remainder term e s z d s z e s 2 has finite mean which is zero unless is noncompact and has null forms further we carry out a numerical investigation of the fermat groups and the results are consistent with an upper bound e s z o e s for all 0 the problem in hyperbolic n space is also investigated

hardy littlewood varieties and semisimple groups we are interested in counting integer and rational points in affine algebraic varieties also under congruence conditions we introduce the notions of a strongly hardy littlewood variety and a relatively hardy littlewood variety in terms of counting rational points satisfying congruence conditions the definition of a strongly hardy littlewood variety is given in such a way that varieties for which the hardy littlewood circle method is applicable are strongly hardy littlewood

a comparison of eigenvalue condition numbers for matrix polynomials abstract in this paper we consider the different condition numbers for simple eigenvalues of matrix polynomials used in the literature and we compare them one of these condition numbers is a generalization of the wilkinson condition number for the standard eigenvalue problem this number has the disadvantage of only being defined for finite eigenvalues in order to give a unified approach to all the eigenvalues of a matrix polynomial both finite and infinite two homogeneous condition numbers have been defined in the literature in their definition very different approaches are used one of the main goals of this note is to show that when the matrix polynomial has a moderate degree both homogeneous condition numbers are essentially the same and one of them provides a geometric interpretation of the other we also show how the homogeneous condition numbers compare with the wilkinson like eigenvalue condition number and how they extend this condition number to zero and infinite eigenvalues

newton s method for overdetermined systems of equations complexity theoretic aspects of continuation methods for the solution of square or underdetermined systems of polynomial equations have been studied by various authors in this paper we consider overdetermined systems where there are more equations than unknowns we study newton s method for such a system

ground state of an infinite two dimensional system of dipoles on a lattice with arbitrary rhombicity angle abstract a class of possible periodic orientational configurations of a system of dipole moments located at the sites of an infinite flat rhombic lattice with an arbitrary rhombicity angle is studied by using the luttinger and tisza method in the framework of the method it is obtained that for 80 the ground state is ferromagnetic at 60 being continuously degenerate in direction for 80 90 the ground state is antiferromagnetic at 90 being also continuously degenerate with respect to one parameter it is shown that the restriction of the interaction range leads to a change in the type of the ground state at 60 this takes place between the third and the fourth coordination spheres and at 30 at a distance of about 500 lattice constants from comparison with known results of numerical experiments on finite systems it is established that the type of the ground state of a finite and an infinite system may be essentially different

the asymptotic distribution of lattice points in euclidean and non euclidean spaces the counting numbers for discrete subgroups of motions in euclidean and non euclidean spaces are obtained using the wave equation as the principal tool in dimensions 2 and 3 the error estimates are close to the best known

classical dipoles on a finite triangular lattice the spherical model approximation abstract in the framework of the spherical model it is shown that a phase transition at finite temperature to a ferromagnetic phase in a two dimensional system with dipole dipole and exchange interactions on a lattice with arbitrary rhombicity angle exists if the ferromagnetic ground state is not continuously degenerate in spin direction when the degeneracy exists the system exhibits essential singularities in the thermodynamic functions at t 0 finite size effects in such systems are investigated in the example of the triangular latice under periodic boundary conditions the existence of only one relevant variable determining the crossover from infinite to finite size behavior is found as t 0 is approached despite that there are two characteristic lengths determining the scale of variation of the correlation function the validity of the general finite size scalling relations in terms of this variable is established