Computational experiences on the distances of polynomials to irreducible polynomials

In this paper we deal with a problem of Turán concerning the `distance' of polynomials to irreducible polynomials. Using computational methods we prove that for any monic polynomial of degree there exists a monic polynomial with deg() = deg() such that is irreducible over and the `distance' of and is .

     

A search for Wieferich and Wilson primes

An odd prime is called a Wieferich prime if

alternatively, a Wilson prime if

To date, the only known Wieferich primes are and , while the only known Wilson primes are , and . We report that there exist no new Wieferich primes , and no new Wilson primes . It is elementary that both defining congruences above hold merely (mod ), and it is sometimes estimated on heuristic grounds that the ``probability" that is Wieferich (independently: that is Wilson) is about . We provide some statistical data relevant to occurrences of small values of the pertinent Fermat and Wilson quotients (mod ).

     

A stochastic particle method for the McKean-Vlasov and the Burgers equation

In this paper we introduce and analyze a stochastic particle method for the McKean-Vlasov and the Burgers equation; the construction and error analysis are based upon the theory of the propagation of chaos for interacting particle systems. Our objective is three-fold. First, we consider a McKean-Vlasov equation in with sufficiently smooth kernels, and the PDEs giving the distribution function and the density of the measure , the solution to the McKean-Vlasov equation. The simulation of the stochastic system with particles provides a discrete measure which approximates for each time (where is a discretization step of the time interval ). An integration (resp. smoothing) of this discrete measure provides approximations of the distribution function (resp. density) of . We show that the convergence rate is for the approximation in of the cumulative distribution function at time , and of order for the approximation in of the density at time ( is the underlying probability space, is a smoothing parameter). Our second objective is to show that our particle method can be modified to solve the Burgers equation with a nonmonotonic initial condition, without modifying the convergence rate . This part extends earlier work of ours, where we have limited ourselves to monotonic initial conditions. Finally, we present numerical experiments which confirm our theoretical estimates and illustrate the numerical efficiency of the method when the viscosity coefficient is very small.

     

A continuity property of multivariate Lagrange interpolation

Let be a sequence of interpolation schemes in of degree (i.e. for each one has unique interpolation by a polynomial of total degree and total order . Suppose that the points of tend to as and the Lagrange-Hermite interpolants, , satisfy for all monomials with . Theorem: for all functions of class in a neighborhood of . (Here denotes the Taylor series of at 0 to order .) Specific examples are given to show the optimality of this result.

     

On solving relative norm equations in algebraic number fields

Let be algebraic number fields and a free -module. We prove a theorem which enables us to determine whether a given relative norm equation of the form has any solutions at all and, if so, to compute a complete set of nonassociate solutions. Finally we formulate an algorithm using this theorem, consider its algebraic complexity and give some examples.

     

Bounds for Multiplicative Cosets over Fields of Prime Order

Let be a positive integer and suppose that is an odd prime with . Suppose that and consider the polynomial . If this polynomial has any roots in , where the coset representatives for are taken to be all integers with , then these roots will form a coset of the multiplicative subgroup of consisting of the th roots of unity mod . Let be a coset of in , and define . In the paper ``Numbers Having Small th Roots mod ' (Mathematics of Computation, Vol. 61, No. 203 (1993),pp. 393-413), Robinson gives upper bounds for of the form , where is the Euler phi-function. This paper gives lower bounds that are of the same form, and seeks to sharpen the constants in the upper bounds of Robinson. The upper bounds of Robinson are proven to be optimal when is a power of or when

     

Construction of high-rank elliptic curves with a nontrivial torsion point

We construct a family of infinitely many elliptic curves over with a nontrivial rational 2-torsion point and with rank 6, which is parametrized by the rational points of an elliptic curve of rank 1.

     

On Wendt's determinant

Wendt's determinant of order is the circulant determinant whose -th entry is the binomial coefficient , for . We give a formula for , when is even not divisible by 6, in terms of the discriminant of a polynomial , with rational coefficients, associated to . In particular, when where is a prime , this yields a factorization of involving a Fermat quotient, a power of and the 6-th power of an integer.

     

Stieltjes polynomials and Lagrange interpolation

Bounds are proved for the Stieltjes polynomial , and lower bounds are proved for the distances of consecutive zeros of the Stieltjes polynomials and the Legendre polynomials . This sharpens a known interlacing result of Szegö. As a byproduct, bounds are obtained for the Geronimus polynomials . Applying these results, convergence theorems are proved for the Lagrange interpolation process with respect to the zeros of , and for the extended Lagrange interpolation process with respect to the zeros of in the uniform and weighted norms. The corresponding Lebesgue constants are of optimal order.

     

Optimal information for approximating periodic analytic functions

Let be a strip in the complex plane. For fixed integer let denote the class of -periodic functions , which are analytic in and satisfy in . Denote by the subset of functions from that are real-valued on the real axis. Given a function , we try to recover at a fixed point by an algorithm on the basis of the information

where , are the Fourier coefficients of . We find the intrinsic error of recovery

Furthermore the -dimensional optimal information error, optimal sampling error and -widths of in , the space of continuous functions on , are determined. The optimal sampling error turns out to be strictly greater than the optimal information error. Finally the same problems are investigated for the class , consisting of all -periodic functions, which are analytic in with -integrable boundary values. In the case sampling fails to yield optimal information as well in odd as in even dimensions.

     

The exponent of discrepancy is at most

We study discrepancy with arbitrary weights in the norm over the -dimensional unit cube. The exponent of discrepancy is defined as the smallest for which there exists a positive number such that for all and all there exist points with discrepancy at most . It is well known that . We improve the upper bound by showing that

This is done by using relations between discrepancy and integration in the average case setting with the Wiener sheet measure. Our proof is not constructive. The known constructive bound on the exponent is .

     

On searching for solutions of the Diophantine equation

We propose a new search algorithm to solve the equation for a fixed value of . By parametrizing min, this algorithm obtains and (if they exist) by solving a quadratic equation derived from divisors of . By using several efficient number-theoretic sieves, the new algorithm is much faster on average than previous straightforward algorithms. We performed a computer search for 51 values of below 1000 (except ) for which no solution has previously been found. We found eight new integer solutions for and in the range of .

     

On a Problem of Byrnes concerning Polynomials with Restricted Coefficients

We consider a question of Byrnes concerning the minimal degree of a polynomial with all coefficients in which has a zero of a given order at . For , we prove his conjecture that the monic polynomial of this type of minimal degree is given by , but we disprove this for . We prove that a polynomial of this type must have , which is in sharp contrast with the situation when one allows coefficients in . The proofs use simple number theoretic ideas and depend ultimately on the fact that .

     

On the asymptotic spectrum of Hermitian block Toeplitz matrices with Toeplitz blocks

We study the asymptotic behaviour of the eigenvalues of Hermitian block Toeplitz matrices , with Toeplitz blocks. Such matrices are generated by the Fourier coefficients of an integrable bivariate function , and we study their eigenvalues for large and , relating their behaviour to some properties of as a function; in particular we show that, for any fixed , the first eigenvalues of tend to , while the last tend to , so extending to the block case a well-known result due to Szegö. In the case the 's are positive-definite, we study the asymptotic spectrum of , where is a block Toeplitz preconditioner for the conjugate gradient method, applied to solve the system , obtaining strict estimates, when and are fixed, and exact limit values, when and tend to infinity, for both the condition number and the conjugate gradient convergence factor of the previous matrices. Extensions to the case of a deeper nesting level of the block structure are also discussed.

     

Accelerated polynomial approximation of finite order entire functions by growth reduction

Let be an entire function of positive order and finite type. The subject of this note is the convergence acceleration of polynomial approximants of by incorporating information about the growth of for . We consider ``near polynomial approximation' on a compact plane set , which should be thought of as a circle or a real interval. Our aim is to find sequences of functions which are the product of a polynomial of degree and an ``easy computable' second factor and such that converges essentially faster to on than the sequence of best approximating polynomials of degree . The resulting method, which we call Reduced Growth method (-method) is introduced in Section 2. In Section 5, numerical examples of the -method applied to the complex error function and to Bessel functions are given.

     

New estimates for Ritz vectors

The following estimate for the Rayleigh-Ritz method is proved:

Here is a bounded self-adjoint operator in a real Hilbert/euclidian space, one of its eigenpairs, a trial subspace for the Rayleigh-Ritz method, and a Ritz pair. This inequality makes it possible to analyze the fine structure of the error of the Rayleigh-Ritz method, in particular, it shows that if an eigenvector is close to the trial subspace with accuracy and a Ritz vector is an approximation to another eigenvector, with a different eigenvalue. Generalizations of the estimate to the cases of eigenspaces and invariant subspaces are suggested, and estimates of approximation of eigenspaces and invariant subspaces are proved.

     

Pure product polynomials and the Prouhet-Tarry-Escott problem

An -factor pure product is a polynomial which can be expressed in the form for some natural numbers . We define the norm of a polynomial to be the sum of the absolute values of the coefficients. It is known that every -factor pure product has norm at least . We describe three algorithms for determining the least norm an -factor pure product can have. We report results of our computations using one of these algorithms which include the result that every -factor pure product has norm strictly greater than if is , , , or .

     

Computing Canonical Heights With Little (or No) Factorization

Let be an elliptic curve with discriminant , and let . The standard method for computing the canonical height is as a sum of local heights . There are well-known series for computing the archimedean height , and the non-archimedean heights are easily computed as soon as all prime factors of have been determined. However, for curves with large coefficients it may be difficult or impossible to factor . In this note we give a method for computing the non-archimedean contribution to which is quite practical and requires little or no factorization. We also give some numerical examples illustrating the algorithm.

     

Approximating the number of integers free of large prime factors

Define to be the number of positive integers such that has no prime divisor larger than . We present a simple algorithm that approximates in floating point operations. This algorithm is based directly on a theorem of Hildebrand and Tenenbaum. We also present data which indicate that this algorithm is more accurate in practice than other known approximations, including the well-known approximation , where is Dickman's function.

     

A quasi-optimal error estimate for a discrete singularly perturbed approximation to the prescribed curvature problem

Solutions of the so-called prescribed curvature problem
, being the curvature field, are approximated via a singularly perturbed elliptic PDE of bistable type. For nondegenerate relative minimizers we prove an error estimate (where stands for the perturbation parameter), and show that this estimate is quasi-optimal. The proof is based on the construction of accurate barriers suggested by formal asymptotics. This analysis is next extended to a finite element discretization of the PDE to prove the same error estimate for discrete minima.