random polynomials having few or no real zeros consider a polynomial of large degree n whose coefficients are independent identically distributed nondegenerate random variables having zero mean and finite moments of all orders we show that such a polynomial has exactly k real zeros with probability n b o 1 as n through integers of the same parity as the fixed integer k 0 in particular the probability that a random polynomial of large even degree n has no real zeros is n b o 1 the finite positive constant b is characterized via the centered stationary gaussian process of correlation function sech t 2 the value of b depends neither on k nor upon the specific law of the coefficients under an extra smoothness assumption about the law of the coefficients with probability n b o 1 one may specify also the approximate locations of the k zeros on the real line the constant b is replaced by b 2 in case the i i d coefficients have a nonzero mean

asymptotic limits of the full navier stokes fourier poisson system abstract the combined quasineutral and zero electron mass limit of the full navier stokes fourier poisson system in the torus t d d 1 is considered it is showed that for well prepared initial data the weak solutions of the full navier stokes fourier poisson system converge to the strong solution of the incompressible navier stokes equations as long as the latter exists moreover the convergence rates are obtained

complex roots of a class of random algebraic polynomials abstract the average density of the distribution of complex roots of equation 0 g 0 1 g 1 z 2 g 2 z 2 n 1 g n 1 z n 1 k with random coefficients j and for z c g j 1 j 0 1 n 1 and k a complex constant with equal real and imaginary parts is known the present paper provides a formula for such an average density for any constant real g j and j a j b j where the a j s and b j s j 0 1 n 1 are real standard normal independent random variables and k k 1 ik 2 where k 1 and k 2 are constants independent of z the limiting behaviour of this density function as n tends to infinity for several special selected g j s is obtained

correlations between zeros of a random polynomial we obtain exact analytical expressions for correlations between real zeros of the kac random polynomial we show that the zeros in the interval 1 1 are asymptotically independent of the zeros outside of this interval and that the straightened zeros have the same limit translation invariant correlations then we calculate the correlations between the straightened zeros of theo 1 random polynomial

universal fluctuations of zeros of chaotic wavefunctions wavefunctions of one and two dimensional quantum systems can be parametrized by a finite number of zeros lying in phase space we study correlations of these zeros for fully chaotic systems in terms of a statistical model based on random polynomials excellent agreement is found for the two point correlation function and nearest neighbour spacing distribution of this model and the results obtained for wavefunctions of dynamical systems we conjecture that these correlation functions are valid for any chaotic system after rescaling the phase space distances unfolding some consequences for the distribution of zeros due to time reversal symmetry are also discussed

optimal reconstruction of images from localized phase the importance of localized phase in signal representation is investigated the convergence rate of the pocs algorithm projection onto convex sets used for image reconstruction from spectral phase is defined and analyzed and the characteristics of images optimally reconstructed from phase only information are presented it is concluded that images of geometric form are most efficiently reconstructed from their spectral phase whereas images of symmetric form have the poorest convergence characteristics the transition between the two extremes is shown to be continuous the results provide a new approach and analysis of the previously reported advantages of the localized phase representation over the global approach and suggest possible compression schemes

the zero electron mass limit in the hydrodynamic model for plasmas abstract the limit of the vanishing ratio of the electron mass to the ion mass in the isentropic transient euler poisson equations with periodic boundary conditions is proved the equations consist of the balance laws for the electron density and current density for a given ion density coupled to the poisson equation for the electrostatic potential the limit is related to the low mach number limit of klainerman and majda in particular the limit velocity satisfies the incompressible euler equations with damping the difference to the zero mach number limit comes from the electrostatic potential which needs to be controlled this is done by a reformulation of the equations in terms of the enthalpy higher order energy estimates and a careful use of the poisson equation

chaos revealing multiplicative representation of quantum eigenstates the quantisation of the two dimensional toric and spherical phase spaces is considered in analytic coherent state representations every pure quantum state admits therein a finite multiplicative parametrisation by the zeros of its husimi function for eigenstates of quantised systems this description explicitly reflects the nature of the underlying classical dynamics in the semiclassical regime the distribution of the zeros in the phase space becomes one dimensional for integrable systems and highly spread out conceivably uniform for chaotic systems this multiplicative representation thereby acquires a special relevance for semiclassical analysis in chaotic systems

quasi neutral and zero viscosity limits of navier stokes poisson equations in the half space abstract the present paper is concerned with the quasi neutral and zero viscosity limits of navier stokes poisson equations in the half space we consider the navier slip boundary condition for velocity and dirichlet boundary condition for electric potential by means of asymptotic analysis with multiple scales we construct an approximate solution of the navier stokes poisson equations involving two different kinds of boundary layer and establish the linear stability of the boundary layer approximations by conormal energy estimate

the berry phase for spin in the majorana representation a continuous cyclic sequence of quantum states has an associated geometric or berry phase for spin j such a sequence is described by a cyclic change in the 2j 1 coefficients of the basis states the berry phase is analysed here for the general case that is the coefficients are allowed to vary in an arbitrary cyclic manner the result is expressed in geometric terms specifically in the democratic representation due to majorana this uniquely characterizes the spin state up to overall phase by the positions of 2j dots on the unit sphere of directions in real space if the positions are denoted by unit vectors where each traces out a parametrized loop on the sphere and the berry phase is given by an integral of combinations of these vectors

correlations between zeros and supersymmetry in our previous work bsz2 we proved that the correlation functions for simultaneous zeros of random generalized polynomials have universal scaling limits and we gave explicit formulas for pair correlations in codimensions 1 and 2 the purpose of this paper is to compute these universal limits in all dimensions and codimensions first we use a supersymmetry method to express the n point correlations as berezin integrals then we use the wick method to give a closed formula for the limit pair correlation function for the point case in all dimensions

on the zeros of the husimi distribution the basic features of the zeros of the husimi phase space density of a quantum eigenstate are discussed it is demonstrated that some special properties of the harmonic oscillator related to the number of zeros and their location within the classical energy shell are not valid for anharmonic potentials as illustrated by numerical examples for the morse oscillator also discussed is the distribution of zeros in quantum poincare sections for systems with two degrees of freedom

time reversal symmetry and random polynomials we analyse the density of roots of random polynomials where each complex coefficient is constructed of a random modulus and a fixed deterministic phase the density of roots is shown to possess a singular component only in the case for which the phases increase linearly with the index of coefficients this means that contrary to earlier belief eigenvectors of a typical quantum chaotic system with some antiunitary symmetry will not display a clustering curve in the stellar representation moreover a class of time reverse invariant quantum systems is shown for which spectra display fluctuations characteristic of orthogonal ensemble while eigenvectors confer to predictions of unitary ensemble

the complex zeros of random polynomials mark kac gave an explicit formula for the expectation of the number vn a of zeros of a random polynomial n i pn z e tj j o in any measurablc subset q of the reals here in i are independent standard normal random variables in fact for each n 1 he obtained an explicit intensity function gn for which e vn l jgn x dx here we extend this formula to obtain an explicit formula for the expected number of zeros in any measurable subset q of the complex plane c namely we show that e vn ki j hn x y dxdy j gn x dx where hn is an explicit intensity function we also study the asymptotics of hn showing that for large n its mass lies close to and is uniformly distributed around the unit circle

distribution of zeros of random and quantum chaotic sections of positive line bundles we study the limit distribution of zeros of certain sequences of holomorphic sections of high powers m n of a positive holomorphic hermitian line bundle l over a compact complex manifold m our first result concerns random sequences of sections using the natural probability measure on the space of sequences of orthonormal bases s n j of h 0 m l n we show that for almost every sequence s n j the associated sequence of zero currents 1 n z s n j tends to the curvature form of l thus the zeros of a sequence of sections s n h 0 m l n chosen independently and at random become uniformly distributed our second result concerns the zeros of quantum ergodic eigenfunctions where the relevant orthonormal bases s n j of h 0 m l n consist of eigensections of a quantum ergodic map we show that also in this case the zeros become uniformly distributed

density of complex critical points of a real random so m 1 polynomial we study the density of complex critical points of a real random so m 1 polynomial in m variables in a previous paper macdonald in j stat phys 136 5 807 2009 the author used the poincare lelong formula to show that the density of complex zeros of a system of these real random polynomials rapidly approaches the density of complex zeros of a system of the corresponding complex random polynomials the su m 1 polynomials in this paper we use the kac rice formula to prove an analogous result the density of complex critical points of one of these real random polynomials rapidly approaches the density of complex critical points of the corresponding complex random polynomial in one variable we give an exact formula and a scaling limit formula for the density of critical points of the real random so 2 polynomial as well as for the density of critical points of the corresponding complex random su 2 polynomial

truncations of random unitary matrices we analyse properties of non hermitian matrices of size m constructed as square submatrices of unitary orthogonal random matrices of size n m distributed according to the haar measure in this way we define ensembles of random matrices and study the statistical properties of the spectrum located inside the unit circle in the limit of large matrices this ensemble is characterized by the ratio m n for the truncated cue we analytically derive the joint density of eigenvalues and all correlation functions in the strongly non unitary case universal ginibre behaviour is found for n m fixed and n to the universal resonance width distribution with n m open channels is recovered

the zeros of random polynomials further results and applications we consider the density of the zeros of random polynomials with nonzero mean correlated gaussian coefficients we show that for most communication problems the original result by hammersley 1956 can be expressed in a simpler form in order to illustrate the usefulness of the presented theory two applications are considered first the zeros of frequency selective ricean fading channels are investigated second the zeros of the transfer function corresponding to channel impulse response estimates obtained by a least squares approach are calculated for both applications implications for equalizer design are discussed

secular determinants of random unitary matrices we consider the characteristic polynomials of random unitary matrices u drawn from various circular ensembles in particular the statistics of the coefficients of these polynomials are studied the variances of these secular coefficients are given explicitly for arbitrary dimension and continued analytically to arbitrary values of the level repulsion exponent the latter secular coefficients are related to the traces of powers of u by newton s well known formulae while the traces tend to have gaussian distributions and to be statistically independent among one another in the limit as the matrix dimension grows large the secular coefficients exhibit strong mutual correlations due to newton s mixing of traces to coefficients these results might become relevant for current efforts at combining semiclassics and random matrix theory in quantum treatments of classically chaotic dynamics

weak coherent states related to the multiplicative group mathbb c generalized coherent states which are associated with the multiplicative group of non zero complex numbers are introduced they have the property of temporal stability and they form a total set in some subspace of the full hilbert space they are highly non classical states in the sense that in general their bargmann functions have zeros which are related to regions where their wigner functions take negative values states with bargmann functions sin az and exp bz az k are examples of the general formalism and they are discussed in detail

random analytic chaotic eigenstates the statistical properties of random analytic functions z are investigated as a phase space model for eigenfunctions of fully chaotic systems we generalize to the plane and to the hyperbolic plane a theorem concerning the equidistribution of the zeros of z previously demonstrated for a spherical phase space su 2 polynomials for systems with time reversal symmetry the number of real roots is computed for the three geometries in the semiclassical regime the local correlation functions are shown to be universal independent of the system considered or the geometry of phase space in particular the autocorrelation function of is given by a gaussian function the connections between this model and the gaussian random function hypothesis as well as the random matrix theory are discussed

phase space approach to quantum dynamics replaces the schrodinger equation for the time propagation of states of a quantized 2d spherical phase space by the dynamics of a system of n particles lying in phase space this is done through factorization formulae of an analytic function theory arising in a coherent state representation the particles being the zeroes of the quantum state for linear hamiltonians like a spin in a uniform magnetic field the motion of the particles is classical however nonlinear terms induce interactions between the particles their time propagation is studied and it is shown that contrary to integrable systems for chaotic maps they tend to fill as their classical counterpart the whole phase space in a uniform way

poincare lelong approach to universality and scaling of correlations between zeros this note is concerned with the scaling limit as n of n point correlations between zeros of random holomorphic polynomials of degree n in m variables more generally we study correlations between zeros of holomorphic sections of powers ln of any positive holomorphic line bundle l over a compact kahler manifold distances are rescaled so that the average density of zeros is independent of n our main result is that the scaling limits of the correlation functions and more generally of the correlation forms are universal i e independent of the bundle l manifold m or point on m

a generalisation of the relation between zeros of the complex kac polynomial and eigenvalues of truncated unitary matrices the zeros of the random laurent series 1 mu sum j 1 infty c j z j where each c j is an independent standard complex gaussian is known to correspond to the scaled eigenvalues of a particular additive rank 1 perturbation of a standard complex gaussian matrix for the corresponding random maclaurin series obtained by the replacement z mapsto 1 z we show that these same zeros correspond to the scaled eigenvalues of a particular multiplicative rank 1 perturbation of a random unitary matrix since the correlation functions of the latter are known by taking an appropriate limit the correlation functions for the random maclaurin series can be determined only for mu rightarrow infty is a determinantal point process obtained for the one and two point correlations by regarding the maclaurin series as the limit of a random polynomial a direct calculation can also be given

the quasineutral limit of compressible navier stokes poisson system with heat conductivity and general initial data abstract the quasineutral limit of compressible navier stokes poisson system with heat conductivity and general ill prepared initial data is rigorously proved in this paper it is proved that as the debye length tends to zero the solution of the compressible navier stokes poisson system converges strongly to the strong solution of the incompressible navier stokes equations plus a term of fast singular oscillating gradient vector fields moreover if the debye length the viscosity coefficients and the heat conductivity coefficient independently go to zero we obtain the incompressible euler equations in both cases the convergence rates are obtained

chaotic analytic zero points exact statistics for those of a random spin state a natural statistical ensemble of 2j points on the unit sphere can be associated via the majorana representation with a random quantum state of spin j and an exact expression is obtained here for the general k point correlation function in this ensemble the pair correlation in the large j limit takes the relatively simple form where and is the angular separation of the pair of points on the sphere it appears from the numerical work of others that in this limit these statistics are typical of the zero points of analytic functions associated with chaotic quantum dynamical systems

distribution of roots of random polynomials we consider polynomials of high degree with random coefficients which appear in the context of quantum chaotic dynamics and investigate various conditions under which their roots tend to concentrate near the unit circle in the complex plane correlation functions of roots are computed analytically we also investigate a certain class of random polynomials whose roots cover in a uniform way the riemann sphere special emphasis is devoted to the influence of symmmetries

hole probability for zeroes of gaussian taylor series with finite radii of convergence we study a family of random taylor series begin aligned f z sum n ge 0 zeta n a n z n end aligned with radius of convergence almost surely 1 and independent identically distributed complex gaussian coefficients zeta n these taylor series are distinguished by the invariance of their zero sets with respect to isometries of the unit disk we find reasonably tight upper and lower bounds on the probability that f does not vanish in the disk z leqslant r as r uparrow 1 our bounds take different forms according to whether the non random coefficients a n grow decay or remain of the same order the results apply more generally to a class of gaussian taylor series whose coefficients a n display power law behavior

an infinite pebble game and applications we generalize the pebble game to infinite directed acyclic graphs and use this generalization to give new and shorter proofs of the following well known results 1 that unbounded memory increases the power of logics of programs and 2 that there exists a context free grammar with infinite index

parametric statistics of zeros of husimi representations of quantum chaotic eigenstates and random polynomials local parametric statistics of zeros of husimi representations of quantum eigenstates are introduced it is conjectured that for a classically fully chaotic system one should use the model of parametric statistics of complex roots of gaussian random polynomials which is exactly solvable as demonstrated below for example the velocities derivatives of zeros of husimi function with respect to an external parameter are predicted to obey a universal non maxwellian distribution where is the mean square velocity the conjecture is demonstrated numerically in a generic chaotic system with two degrees of freedom dynamical formulation of the zero flow in terms of an integrable many body dynamical system is given as well

su 1 1 random polynomials we study statistical properties of zeros of random polynomials and random analytic functions associated with the pseudoeuclidean group of symmetries su 1 1 by utilizing both analytical and numerical techniques we first show that zeros of the su 1 1 random polynomial of degree n are concentrated in a narrow annulus of the order of n 1 around the unit circle on the complex plane and we find an explicit formula for the scaled density of the zeros distribution along the radius in the limit n our results are supported through various numerical simulations we then extend results of hannay 1 and bleher et al 2 to derive different formulae for correlations between zeros of the su 1 1 random analytic functions by applying the generalized kac rice formula we express the correlation functions in terms of some gaussian integrals which can be evaluated combinatorially as a finite sum over feynman diagrams or as a supersymmetric integral due to the su 1 1 symmetry the correlation functions depend only on the hyperbolic distances between the points on the unit disk and we obtain an explicit formula for the two point correlation function it displays quadratic repulsion at small distances and fast decay of correlations at infinity in an appendix to the paper we evaluate correlations between the outer zeros zj 1 of the su 1 1 random polynomial and we prove that the inner and outer zeros are independent in the limit when the degree of the polynomial goes to infinity

nodal domains statistics a criterion for quantum chaos we consider the distribution of the properly normalized numbers of nodal domains of wave functions in 2d quantum billiards we show that these distributions distinguish clearly between systems with integrable separable or chaotic underlying classical dynamics and for each case the limiting distribution is universal system independent thus a new criterion for quantum chaos is provided by the statistics of the wave functions which complements the well established criterion based on spectral statistics

quasi neutral limit of the full navier stokes fourier poisson system abstract the quasi neutral limit of the full navier stokes fourier poisson system in the torus t d d 1 is considered we rigorously prove that as the scaled debye length goes to zero the global in time weak solutions of the full navier stokes fourier poisson system converge to the strong solution of the incompressible navier stokes equations as long as the latter exists in particular the effect of large temperature variations is taken into account

roots of random polynomials whose coefficients have logarithmic tails it has been shown by ibragimov and zaporozhets in prokhorov and contemporary probability theory 2013 springer that the complex roots of a random polynomial gn z nk 0 kzk with i i d coefficients 0 n concentrate a s near the unit circle as n if and only if elog 0 we study the transition from concentration to deconcentration of roots by considering coefficients with tails behaving like l log t log t as t where 0 and l is a slowly varying function under this assumption the structure of complex and real roots of gn is described in terms of the least concave majorant of the poisson point process on 0 1 0 with intensity v 1 dudv

exact statistics of complex zeros for gaussian random polynomials with real coefficients k point correlations of complex zeros for gaussian ensembles of random polynomials of order n with real coefficients grprc are calculated exactly following an approach of hannay for the case of gaussian random polynomials with complex coefficients grpcc it is shown that in the thermodynamic limit of gaussian random holomorphic functions all the statistics converge to their grpcc counterparts as one moves off the real axis while close to the real axis the two cases are essentially different special emphasis is given to one and two point correlation functions in various regimes

equidistribution of zeros of holomorphic sections in the non compact setting we consider tensor powers l n of a positive hermitian line bundle l h l over a non compact complex manifold x in the compact case b shiffman and s zelditch proved that the zeros of random sections become asymptotically uniformly distributed as n with respect to the natural measure coming from the curvature of l under certain boundedness assumptions on the curvature of the canonical line bundle of x and on the chern form of l we prove a non compact version of this result we give various applications including the limiting distribution of zeros of cusp forms with respect to the principal congruence subgroups of sl 2 and to the hyperbolic measure the higher dimensional case of arithmetic quotients and the case of orthogonal polynomials with weights at infinity we also give estimates for the speed of convergence of the currents of integration on the zero divisors

distribution of roots of random real generalized polynomials the average density of zeros for monic generalized polynomials p n z phi z sum nolimits k 1 n c k f k z with real holomorphic f k and real gaussian coefficients is expressed in terms of correlation functions of the values of the polynomial and its derivative we obtain compact expressions for both the regular component generated by the complex roots and the singular one real roots of the average density of roots the density of the regular component goes to zero in the vicinity of the real axis like lmz we present the low and high disorder asymptotic behaviors then we particularize to the large n limit of the average density of complex roots of monic algebraic polynomials of the form p n z z n sum nolimits k 1 n c k z n k with real independent identically distributed gaussian coefficients having zero mean and dispersion delta 1 sqrt n lambda the average density tends to a simple universal function of 2nlog z and in the domain coth 2 n sin arg z where nearly all the roots are located for largen

unbounded program memory adds to the expressive power of first order programming logic it is proved that no logic of programs with unbounded memory is reducible to a bounded memory programming logic this is achieved by carefully analyzing firstorder definability in the algebra of finite binary trees

random riesz energies on compact kahler manifolds the expected riesz energies e n h es of the zero sets of systems of independent gaussian random polynomials are determined asymptotically as the degree n in all dimensions and codimensions the asymptotics are proved for sections of any positive line bundle over any compact kahler manifold

quasi neutral limit of a nonlinear drift diffusion model for semiconductors the limit of the vanishing debye length the charge neutral limit in a nonlinear bipolar drift diffusion model for semiconductors without a pn junction i e with a unipolar background charge is studied the quasi neutral limit zero debye length limit is determined rigorously by using the so called entropy functional which yields appropriate uniform estimates

definability by programs in first order structures introduction ti 1 preliminarv detinitilms and results algebraic characterizations of the unwind property a structure with the unwind property for every iterative but not every recursive 1 program the truth table property structures with the truth table property but not the unwind property for every iterative program 4ppendix 1 the pebble game on infinite dags ippendix 2 remaining proofs references

complex roots of a random algebraic polynomial abstract this paper for any constantk provides an exact formula for the average density of the distribution of the complex roots of equation 0 1z 2z2 n 1zn 1 kwhere j aj ibjand aj n 1j 0and bj n 1j 0are sequences of independent identically and normally distributed random variables andkis a complex number withkas its real and imaginary parts the case of real roots of the above equation with real coefficients andk z ris well known further we obtain the limiting behaviour of this distribution function asntends to infinity

convergence of the navier stokes poisson system to the incompressible navier stokes equations the quasineutral limit of the navier stokes poisson system in the whole space rd d 1 and in the torus td is studied in this paper it is shown that for the well prepared initial data the global weak solution of the navier stokes poisson system converges strongly to the strong solution of the incompressible navier stokes equations

complex zeros of trigonometric polynomials with standard normal random coefficients abstract in this paper we obtain an exact formula for the average density of the distribution of complex zeros of a random trigonometric polynomial 0 1 cos 2 cos 2 n cos n in 0 2 where the coefficients j a j b j and a j j 1 n and b j j 1 n are sequences of independent normally distributed random variables with mean 0 and variance 1 we also provide the limiting behaviour of the zeros density function as n tends to infinity the corresponding results for the case of random algebraic polynomials are known

on roots of random polynomials we study the distribution of the complex roots of random polynomials of degree n with i i d coefficients using techniques related to rice s treatment of the real roots question we derive under appropriate moment and regularity conditions an exact formula for the average density of this distribution which yields appropriate limit average densities further using a different technique we prove limit distribution results for coefficients in the domain of attraction of the stable law

signed zeros of gaussian vector fields density correlation functions and curvature we calculate correlation functions of the signed density of zeros of gaussian distributed vector fields we are able to express correlation functions of arbitrary order through the curvature tensor of a certain abstract riemann cartan or riemannian manifold as an application we discuss one and two point functions the zeros of a two dimensional gaussian vector field model the distribution of topological defects in the high temperature phase of two dimensional systems with orientational degrees of freedom such as superfluid films thin superconductors and liquid crystals

root statistics of random polynomials with bounded mahler measure the mahler measure of a polynomial is a measure of complexity formed by taking the modulus of the leading coecient times the modulus of the product of its roots outside the unit circle the roots of a real degree n polynomial chosen uniformly from the set of polynomials of mahler measure at most 1 yields a pfaan point process on the complex plane when n is large with probability tending to 1 the roots tend to the unit circle and we investigate the asymptotics of the scaled kernel in a neighborhood of a point on the unit circle when this point is away from the real axis on which there is a positive probability of nding a root the scaled process degenerates to a determinantal point process with the same local statistics i e scalar kernel as the limiting process formed from the roots of complex polynomials chosen uniformly from the set of polynomials of mahler measure at most 1 three new matrix kernels appear in a neighborhood of 1 which encode information about the correlations between real roots between complex roots and between real and complex roots away from the unit circle the kernels converge to new limiting kernels which imply among other things that the expected number of roots in any open subset of c disjoint from the unit circle converges to a positive number we also give ensembles with identical statistics drawn from two dimensional electrostatics with potential theoretic weights and normal matrices chosen with regard to their topological entropy as actions on euclidean space

two point eigenvalue correlation for a class of non selfadjoint operators under random perturbations we consider a non selfadjoint h differential model operator ph in the semiclassical limit h rightarrow 0 subject to random perturbations with a small coupling constant assume that rm e frac 1 ch text 0 suitably large let be the closure of the range of the principal symbol

on the complexity of real root isolation using continued fractions we present algorithmic complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real algebraic numbers one motivation is to explain the method s good performance in practice we derive an expected complexity bound of email protected b d 6 d email protected 2 where d is the polynomial degree and t bounds the coefficient bit size using a standard bound on the expected bit size of the integers in the continued fraction expansion thus matching the current worst case complexity bound for real root isolation by exact methods sturm descartes and bernstein subdivision moreover using a homothetic transformation we improve the expected complexity bound to email protected b d email protected we compute the multiplicities within the same complexity and extend the algorithm to non square free polynomials finally we present an open source c implementation in the algebraic library synaps and illustrate its completeness and efficiency as compared to some other available software for this we use polynomials with coefficient bit size up to 8000 bits and degree up to 1000

real roots of random polynomials universality close to accumulation points we identify the scaling region of a width o n 1 in the vicinity of the accumulation points t 1 of the real roots of a random kac like polynomial of large degree n we argue that the density of the real roots in this region tends to a universal form shared by all polynomials with independent identically distributed coefficients ci as long as the second moment e c2i is finite in particular we reveal a gradual in contrast to the previously reported abrupt and quite nontrivial suppression of the number of real roots for coefficients with a nonzero mean value n e ci scaled as n n 1 2

universal results for correlations of characteristic polynomials riemann hilbert approach we prove that general correlation functions of both ratios and products of characteristic polynomials of hermitian random matrices are governed by integrable kernels of three different types a those constructed from orthogonal polynomials b those constructed from cauchy transforms of the same orthogonal polynomials and finally c those constructed from both orthogonal polynomials and their cauchy transforms these kernels are related with the riemann hilbert problem for orthogonal polynomials for the correlation functions we obtain exact expressions in the form of determinants of these kernels derived representations enable us to study asymptotics of correlation functions of characteristic polynomials via the deift zhou steepest descent stationary phase method for riemann hilbert problems and in particular to find negative moments of characteristic polynomials this reveals the universal parts of the correlation functions and moments of characteristic polynomials for an arbitrary invariant ensemble of 2 symmetry class

universality and scaling of correlations between zeros on complex manifolds we study the limit as n of the correlations between simultaneous zeros of random sections of the powers l n of a positive holomorphic line bundle l over a compact complex manifold m when distances are rescaled so that the average density of zeros is independent of n we show that the limit correlation is independent of the line bundle and depends only on the dimension of m and the codimension of the zero sets we also provide some explicit formulas for pair correlations in particular we prove that hannay s limit pair correlation function for su 2 polynomials holds for all compact riemann surfaces

quantum chaotic dynamics and random polynomials we investigate the distribution of roots of polynomials of high degree with random coefficients which among others appear naturally in the context of quantum chaotic dynamics it is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane in order to further increase this tendency we study in detail the particular case of self inversive random polynomials and show that for them a finite portion of all roots lies exactly on the unit circle correlation functions of these roots are also computed analytically and compared to the correlations of eigenvalues of random matrices the problem of ergodicity of chaotic wavefunctions is also considered for that purpose we introduce a family of random polynomials whose roots spread uniformly over phase space while these results are consistent with random matrix theory predictions they provide a new and different insight into the problem of quantum ergodicity special attention is devoted to the role of symmetries in the distribution of roots of random polynomials

the chaotic analytic function in contrast to a stationary gaussian random function of a real variable which is free to have any correlation function the closest analogous analytic random function in the complex plane has no true freedom it is statistically unique since it has arisen only recently as an apparently universal feature in the physical context of quantum chaos i refer to it here as the chaotic analytic function i note that it is implied by the assumption that a quantum chaotic wavefunction has gaussian randomness and has a constant value for the average of its wigner function in phase space interpreted literally this shows that the chaotic analytic function is the bargmann function of a pure white noise wavefunction more physically if constant is replaced by smooth on the scale of a planck area these assumptions are the semiclassical ones made by berry for chaotic eigenstates the analysis shows that the chaotic analytic function is still obtained semiclassically

large deviations for the fermion point process associated with the exponential kernel for the fermion point process on the whole complex plane associated with the exponential kernel e z bar w we show the central limit theorem for the random variable d r the number of points inside the ball d r of radius r as r and we establish the large deviation principle for the random variables r 2 d r r 0

rate of convergence from the navier stokes poisson system to the incompressible euler equations the rate of convergence from the navier stokes poisson system to the incompressible euler equations is obtained when the combined quasineutral and vanishing viscosity limits is considered

on the distribution of zeros of chaotic wavefunctions wavefunctions in a phase space representation can be expressed as entire functions of their zeros if the phase space is compact these zeros seem to hide a lot of relevant and explicit information about the underlying classical dynamics besides an understanding of their statistical properties may prove to be useful in the analytical calculations of the wavefunctions in quantum chaotic systems this motivates us to pursue the present study which by a numerical statistical analysis shows that both long range correlations as well as short range correlations exist between zeros while the latter turn out to be universal and parametric independent the former seem to be system dependent and are significantly affected by various parameters i e symmetry localization etc furthermore for the delocalized quantum dynamics the distribution of these zeros seem to mimic that of the zeros of the random functions as well as random polynomials

joint distribution of conjugate algebraic numbers a random polynomial approach abstract we count the algebraic numbers of fixed degree by their w weighted l p norm which generalizes the naive height the length the euclidean and the bombieri norms for non negative integers k l such that k 2 l n and a borel subset b r c l denote by p w k l q b the number of ordered k l tuples in b of conjugate algebraic numbers of degree n and w weighted l p norm at most q we show that lim q p w k l q b q n 1 vol n 1 b p w n 1 2 n 1 b p w k l x z d x d z where vol n 1 b p w n 1 is the volume of the unit w weighted l p ball and p w k l will denote the correlation function of k real and l complex zeros of the random polynomial j 1 n j w j z j where j are i i d random variables with density c p e t p for 0 p and with constant density on 1 1 for p if the boundary of b is of lipschitz type we also estimate the rate of convergence we give an explicit formula for p w k l which in the case k 2 l n has a very simple form to this end we obtain a general formula for the correlations between real and complex zeros of a random polynomial with arbitrary independent absolutely continuous coefficients

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

how many zeros of a random polynomial are real we provide an elementary geometric derivation of the kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients we show that the expected number of real zeros is simply the length of the moment curve 1 t tn projected onto the surface of the unit sphere divided by the probability density of the real zeros is proportional to how fast this curve is traced out we then relax kac s assumptions by considering a variety of random sums series and distributions and we also illustrate such ideas as integral geometry and the fubini study metric

zeros of the i i d gaussian power series a conformally invariant determinantal process consider the zero set of the random power series f z p anz n with i i d complex gaussian coefficientsan we show that these zeros form a determinantal process more precisely their joint intensity can be written as a minor of the bergman kernel we show that the number of zeros of f in a disk of radius r about the origin has the same distribution as the sum of independent 0 1 valued random variables xk where p xk 1 r 2k moreover the set of absolute values of the zeros of f has the

scaled correlations of critical points of random sections on riemann surfaces in this paper we prove that as n goes to infinity the scaling limit of the correlation between critical points z1 and z2 of random holomorphic sections of the n th power of a positive line bundle over a compact riemann surface tends to 2 3 2 for small sqrt n z 1 nobreak z 2 the scaling limit is directly calculated using a general form of the kac rice formula and formulas and theorems of pavel bleher bernard shiffman and steve zelditch

intrinsic ordering of quasienergy states for mixed regular chaotic quantum systems zeros of the husimi distribution a method is proposed which allows an intrinsic ordering of the quasienergy states for a time periodic quantum system according to the number of essential zeros of the husimi distribution in phase space the corresponding classical system shows a typical mixed dynamics with coexisting regular and chaotic regions in phase space the ordering of states allows a straightforward comparison with semiclassical quantities moreover a direct connection of the organization of zeros with the underlying classical phase space dynamics is demonstrated

density of complex zeros of a system of real random polynomials we study the density of complex zeros of a system of real random so m 1 polynomials in m variables we show that the density of complex zeros of this random polynomial system with real coefficients rapidly approaches the density of complex zeros in the complex coefficients case we also show that the behavior the scaled density of complex zeros near m of the system of real random polynomials is different in the m 2 case than in the m 1 case the density approaches infinity instead of tending linearly to zero

analysis of oscillations and defect measures for the quasineutral limit in plasma physics we perform a rigorous analysis of the quasineutral limit for a hydrodynamical model of a viscous plasma represented by the navier stokes poisson system in three dimensions we show that as 0 the velocity field u strongly converges towards an incompressible velocity vector field u and the density fluctuation 1 weakly converges to zero in general the limit velocity field cannot be expected to satisfy the incompressible navier stokes equation indeed the presence of high frequency oscillations strongly affects the quadratic nonlinearities and we have to take care of self interacting wave packets we provide a detailed mathematical description of the convergence process by using microlocal defect measures and by developing an explicit correctors analysis moreover we were able to identify an explicit pseudo parabolic pde satisfied by the leading correctors terms our results include all the previous results in the literature in particular we show that the formal limit holds rigorously in the case of well prepared data

saddle points in the chaotic analytic function and ginibre characteristic polynomial comparison is made between the distribution of saddle points in the chaotic analytic function and in the characteristic polynomials of the ginibre ensemble realizing the logarithmic derivative of these infinite polynomials as the electric field of a distribution of coulombic charges at the zeros a simple mean field electrostatic argument shows that the density of saddles minus zeros falls off as 1 z 4 from the origin this behaviour is expected to be general for finite or infinite polynomials with zeros uniformly randomly distributed in the complex plane and which repel

parametric spectral statistics in unitary random matrix ensembles from distribution functions to intra level correlations we establish a general framework to explore parametric statistics of individual energy levels in unitary random matrix ensembles for a generic confinement potential tr w h we i find the joint distribution functions of the eigenvalues of h and h h v for an arbitrary fixed v both for finite matrix size n and in the thermodynamic n limit ii derive many point parametric correlation functions of the two sets of eigenvalues and show that they are naturally parametrized by the eigenvalues of the reactance matrix for scattering off the potential v iii prove the universality of the correlation functions in unitary ensembles with non gaussian non invariant confinement potential tr w h v iv establish a general scheme for exact calculation of level number dependent parametric correlation functions and apply the scheme to the calculation of intra level velocity autocorrelation function and the distribution of parametric level shifts

the monge metric on the sphere and geometry of quantum states topological and geometrical properties of the set of mixed quantum states in the n dimensional hilbert space are analysed assuming that the corresponding classical dynamics takes place on the sphere we use the vector su 2 coherent states and the generalized husimi distributions to define the monge distance between two arbitrary density matrices the monge metric has a simple semiclassical interpretation and induces a non trivial geometry among all pure states the distance from the maximally mixed state proportional to the identity matrix admits the largest value for the coherent states while the delocalized chaotic states are close to this contrasts the geometry induced by the standard trace hilbert schmidt or bures metrics for which the distance from is the same for all pure states we discuss possible physical consequences including unitary time evolution and the process of decoherence we introduce also a simplified monge metric defined in the space of pure quantum states and more suitable for numerical computation

exact statistical properties of the zeros of complex random polynomials the zeros of complex gaussian random polynomials with coefficients such that the density in the underlying complex space is uniform are known to have the same statistical properties as the zeros of the coherent state representation of one dimensional chaotic quantum systems we extend the interpretation of these polynomials by showing that they also arise as the wavefunction for a quantum particle in a magnetic field constructed from a random superposition of states in the lowest landau level a study of the statistical properties of the zeros is undertaken using exact formulae for the one and two point distribution functions attention is focused on the moments of the two point correlation in the bulk the variance of a linear statistic and the asymptotic form of the two point correlation at the boundary a comparison is made with the same quantities for the eigenvalues of complex gaussian random matrices

explicit formulas for the distribution of complex zeros of a family of random sums abstract the present paper provides an explicit formula for the average intensity of the distribution of complex zeros of a family of random sums of the form s n z j 0 n j f j z where z is the complex variable x i y j j i j and j j 0 n and j j 0 n are sequences of standard normal independent random variables and f j j 0 n is a sequence of given analytic functions that are real valued on the real number line in addition the numerical computations of the intensity functions and the empirical distributions for the special cases of random weyl polynomials random taylor polynomials and random truncated fourier cosine series are included as examples

computing algebraic formulas using a constant number of registers it is shown that over an arbitrary ring the functions computed by polynomial size algebraic formulas are also computed by polynomial length algebraic straight line programs that use only three registers this was previously known for boolean formulas d a barrington j comput system sci 38 1989 pp 150 164 which are equivalent to algebraic formulas over the ring gf 2 for formulas over arbitrary rings the result is an improvement over previous methods that require the number of registers to be logarithmic in the size of the formulas in order to obtain polynomial length straight line programs moreover the straight line programs that arise in these constructions have the property that they consist of statements whose actions on the registers are linear and bijective a consequence of this is that the problem of determining the iterated product of n3 times 3 matrices is complete under p projections for algebraic nc 1 also when the ring is gf 2 the programs that arise in the c

chaotic eigenfunctions in phase space we study individual eigenstates of quantized area preserving maps on the 2 torus which are classically chaotic in order to analyze their semiclassical behavior we use the bargmann husimi representations for quantum states as well as their stellar parametrization which encodes states through a minimal set of points in phase space the constellation of zeros of the husimi density we rigorously prove that a semiclassical uniform distribution of husimi densities on the torus entails a similar equidistribution for the corresponding constellations we deduce from this property a universal behavior for the phase patterns of chaotic bargmann eigenfunctions which is reminiscent of the wkb approximation for eigenstates of integrable systems though in a weaker sense in order to obtain more precise information on chaotic eigenconstellations we then model their properties by ensembles of random states generalizing former results on the 2 sphere to the torus geometry this approach yields statistical predictions for the constellations which fit quite well the chaotic data we finally observe that specific dynamical information e g the presence of high peaks like scars in husimi densities can be recovered from the knowledge of a few long wavelength fourier coefficients which therefore appear as valuable order parameters at the level of individual chaotic eigenfunctions