The correlations of finite Desarguesian planes, Part II: The classification (I)

This paper continues the classification of the correlations of planes of odd nonsquare order. Part I (Generalities) – see reference [1]-included introductory definitions and results (Section 1), algebraic preliminaries (Section 2), as well as a discussion of equivalent correlations (Section 3) and of their general properties (Section 4). The classification proper revolves around a special polynomial which can have one, two, or q + 1 zeros, or no zeros at all, and each of these four possibilities leads to different families of correlations. The present article contains Section 5, devoted to the cases in which the correlation is defined by a diagonal matrix (Subsection 5.1) or the polynomial in the preceding paragraph possesses q + 1 zeros (Subsection 5.2), one zero (Subsection 5.3) and two zeros (Subsection 5.4). Subsection 5.5 presents certain results to be used in the subsequent sections.     

The Haar condition and multiplicity of zeros

Summary In this paper the problem is investigated of how to take the (possibly noninteger) multiplicity of zeros into account in the Haar condition for a linear function space on a given interval. Therefore, a distinction is made between regular and singular points of the interval, and a notion of geometric multiplicity, which always is a positive integer, is introduced. It is pointed out that, for regular zeros (i.e., zeros situate at regular points), aq-fold zero (in the sense that its geometric multiplicity equalsq), counts forq distinct zeros in the Haar condition. For singular zeros (i.e., zeros situated at singular points), this geometric multiplicity has to be diminished by some well-determinable integer.     

Schanuel's Conjecture and Algebraic Roots of Exponential Polynomials

In this article, I will prove that assuming Schanuel's conjecture, an exponential polynomial with algebraic coefficients can have only finitely many algebraic roots. Furthermore, this proof demonstrates that there are no unexpected algebraic roots of any such exponential polynomial. This implies a special case of Shapiro's conjecture: if p(x) and q(x) are two exponential polynomials with algebraic coefficients, each involving only one iteration of the exponential map, and they have common factors only of the form exp (g) for some exponential polynomial g, then p and q have only finitely many common zeros.     

Polynomial solutions of q-Heun equation and ultradiscrete limit

We study polynomial-type solutions of the q-Heun equation, which is related with quasi-exact solvability. The condition that the q-Heun equation has a non-zero polynomial-type solution is described by the roots of the spectral polynomial, whose variable is the accessory parameter E. We obtain sufficient conditions that the roots of the spectral polynomial are all real and distinct. We consider the ultradiscrete limit to clarify the roots of the spectral polynomial and the zeros of the polynomial-type solution of the q-Heun equation.     

Applications of branched values to p-adic functional equations on analytic functions

Let \(\mathbb{K}\) be an algebraically closed field of characteristic 0, complete with respect to an ultrametric absolute value. Results on branched values obtained in a previous paper are used to prove that algebraic functional equations of the form g ^q = hf ^q + w have no solution among transcendental entire functions f, g or among unbounded analytic functions inside an open disk, when w is a polynomial or a bounded analytic function and h is a polynomial or an analytic function whose zeros are of order multiple of q. We also show that an analytic function whose zeros are multiple of an integer q inside a disk is the q-th power of another analytic function, provided q is a prime to the residue characteristic.     

To solving multiparameter problems of algebra. 9. The Ψ<Emphasis Type="Italic">F-q</Emphasis> method for factorizing invariant polynomials and its applications

A new method (the ΨF-q method) for computing the invariant polynomials of a q-parameter (q ≥ 1) polynomial matrix F is suggested. Invariant polynomials are computed in factored form, which permits one to analyze the structure of the regular spectrum of the matrix F, to isolate the divisors of each of the invariant polynomials whose zeros belong to the invariant polynomial in question, to find the divisors whose zeros belong to at least two of the neighboring invariant polynomials, and to determine the heredity levels of points of the spectrum for each of the invariant polynomials. Applications of the ΨF-q method to representing a polynomial matrix F(λ) as a product of matrices whose spectra coincide with the zeros of the corresponding divisors of the characteristic polynomial and, in particular, with the zeros of an arbitrary invariant polynomial or its divisors are considered. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 165–173.     

Counting isomorphism classes of pointed hyperelliptic curves of genus 4 over finite fields with odd characteristic

This paper is devoted to computing the number of isomorphism classes of pointed hyperelliptic curves over finite fields. We deal with the genus-4 case and the finite fields are of odd characteristic. The number of isomorphism classes is computed. This number can be represented as a polynomial in q of degree 7, where q is the order of the finite field. The results have applications in the classification problems and in the hyperelliptic curve cryptosystems.     

Expansions for the Fundamental Hermite Interpolation Polynomials in Terms of Chebyshev Polynomials

We obtain explicit expansions of the fundamental Hermite interpolation polynomials in terms of Chebyshev polynomials in the case where the nodes considered are either zeros of the (n + 1)th-degree Chebyshev polynomial or extremum points of the nth-degree Chebyshev polynomial.     

Properties of solutions ofu″+g(t) u 2n−1 =0, II

Summary Part I [7] of this paper appeared in this journal in 1962. In Sections 1, 2 of this paper we consider the equation of the title wheng(t)>0. In Section 1 we examine the strength of the hypothesis of continuity ofg(t) in known theorems. In Section 2 we determine the combinations of boundedness and oscillation that may occur for solutions of this equation. Section 3 is devoted to properties of solutions of the equation wheng(t)<0. In general, differentiability ofg(t) is not required in the paper.     

Discriminants and Functional Equations for Polynomials Orthogonal on the Unit Circle

We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and q-difference equations for these polynomials. A general functional equation is found which allows one to relate the zeros of the orthogonal polynomials to the stationary values of an explicit quasi-energy and implies recurrences on the orthogonal polynomial coefficients. We also evaluate the discriminants and quantized discriminants of polynomials orthogonal on the unit circle.     

Zeros of Racah coefficients and the pell equation

The interface between Racah coefficients and mathematics is reviewed and several unsolved problems pointed out. The specific goal of this investigation is to determine zeros of these coefficients. The general polynomial is given whose set of zeros contains all nontrivial zeros of Racah (6j) coefficients [this polynomial is also given for the Wigner-Clebsch-Gordan (3j) coefficients]. Zeros of weight 1 3j- and 6j-coefficients are known to be related to the solutions of classic Diophantine equations. Here it is shown how solutions of the quadratic Diophantine equation known as Pell's equation are related to weight 2 zeros of 3j- and 6j-coefficients. This relation involves transformations of quadratic forms over the integers, the orbit classification of zeros of Pell's equation, and an algorithm for determining numerically the fundamental solutions of Pell's equation. The symbol manipulation program MACSYMA was used extensively to effect various factorings and transformations and to give a proof.The results of this paper were presented in an invited talk by one of us (JDL) at the NSF-CBMS Regional Conference on Special Functions, Physics and Computer Algebra, May 20–24, 1985, Arizona State University, Tempe, AZ.     

Suboptimal Kronrod extension formulae for numerical quadrature

Summary We consider cases where the Stieltjes polynomial associated with a Gaussian quadrature formula has complex zeros. In such cases a Kronrod extension of the Gaussian rule does not exist. A method is described for modifying the Stieltjes polynomial so that the resulting polynomial has no complex zeros. The modification is performed in such a way that the Kronrod-type extension rule resulting from the addition of then+1 zeros of the modified Stieltjes polynomial to the original knots of the Gaussian rule has only slightly lower degree of precision than normally achieved when the Kronrod extension rule exists. As examples of the use of the method, we present some new formulae extending the classical Gauss-Hermite quadrature rules. We comment on the limited success of the method in extending Gauss-Laguerre rules and show that several modified extensions of the Gauss Gegenbauer formulae exist in cases where the standard Kronrod extension does not.     

The Nullstellensatz for Systems of PDE

Given an ideal I in , the polynomial ring in n-indeterminates, the affine variety of I is the set of common zeros in ⁿ of all the polynomials that belong to I, and the Hilbert Nullstellensatz states that there is a bijective correspondence between these affine varieties and radical ideals of . If, on the other hand, one thinks of a polynomial as a (constant coefficient) partial differential operator, then instead of its zeros in ⁿ, one can consider its zeros, i.e., its homogeneous solutions, in various function and distribution spaces. An advantage of this point of view is that one can then consider not only the zeros of ideals of , but also the zeros of submodules of free modules over (i.e., of systems of PDEs). The question then arises as to what is the analogue here of the Hilbert Nullstellensatz. The answer clearly depends on the function–distribution space in which solutions of PDEs are being located, and this paper considers the case of the classical spaces. This question is related to the more general question of embedding a partial differential system in a (two-sided) complex with minimal homology. This paper also explains how these questions are related to some questions in control theory.     

To solving multiparameter problems of algebra. 2. The method of partial relative factorization and its applications

For a q-parameter (q 2) polynomial matrix of full rank whose regular and singular spectra have no points in common, a method for computing its partial relative factorization into a product of two matrices with disjoint spectra is suggested. One of the factors is regular and is represented as a product of q matrices with disjoint spectra. The spectrum of each of the factors is independent of one of the parameters and forms in the space ^q a cylindrical manifold w.r.t. this parameter. The method is applied to computing zeros of the minimal polynomial with the corresponding eigenvectors. An application of the method to computing a specific basis of the null-space of polynomial solutions of the matrix is considered. Bibliography: 4 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 296, 2003, pp. 89–107.     

Numerical computation of polynomial zeros by means of Aberth's method

An algorithm for computing polynomial zeros, based on Aberth's method, is presented. The starting approximations are chosen by means of a suitable application of Rouché's theorem. More precisely, an integerq 1 and a set of annuliA _i,i=1,...,q, in the complex plane, are determined together with the numberk _i of zeros of the polynomial contained in each annulusA _i. As starting approximations we choosek _i complex numbers lying on a suitable circle contained in the annulusA _i, fori=1,...,q. The computation of Newton's correction is performed in such a way that overflow situations are removed. A suitable stop condition, based on a rigorous backward rounding error analysis, guarantees that the computed approximations are the exact zeros of a nearby polynomial. This implies the backward stability of our algorithm. We provide a Fortran 77 implementation of the algorithm which is robust against overflow and allows us to deal with polynomials of any degree, not necessarily monic, whose zeros and coefficients are representable as floating point numbers. In all the tests performed with more than 1000 polynomials having degrees from 10 up to 25,600 and randomly generated coefficients, the Fortran 77 implementation of our algorithm computed approximations to all the zeros within the relative precision allowed by the classical conditioning theorems with 11.1 average iterations. In the worst case the number of iterations needed has been at most 17. Comparisons with available public domain software and with the algorithm PA16AD of Harwell are performed and show the effectiveness of our approach. A multiprecision implementation in MATHEMATICA is presented together with the results of the numerical tests performed.Work performed under the support of the ESPRIT BRA project 6846 POSSO (POlynomial System SOlving).     

On the chromaticity of certain subgraphs of a q-tree

We show that a graph G on n ? q + 1 vertices (where q ? 2) has the chromatic polynomial P(G;λ) = λ(λ ? 1) … (λ ? q + 2) (λ ? q + 1)² (λ ? q)^n?q?1 if and only if G can be obtained from a q-tree Ton n vertices by deleting an edge contained in exactly q ? 1 triangles of T. Furthermore, we prove that these graphs are triangulated.     

The classification of spreads in PG(3, q) admitting linear groups of order q(q+1), I. odd order

A classification is given of all spreads in PG(3, q), q = p^r, p odd, whose associated translation planes admit linear collineation groups of order q(q +1) such that a Sylow p-subgroup fixes a line and acts non-trivially on it.The authors are indebted to T. Penttila for pointing out the special examples of conical flock translation planes of order q² that admit groups of order q(q+1), when q = 23 or 47.     

Improvements upon the Chevalley-Warning-Ax-Katz-type estimates

Let f be a polynomial over finite field Fq with q elements and let N(f=0) denote the number of zeros of f in Fq. The q-divisibility properties of N(f=0) have been studied by many authors, such as Chavelley, Warning, Ax, Katz, etc. In this paper, by reducing the degree of a given polynomial meanwhile remaining the number of its zeros unchanged, we present an improvement upon the Chevalley-Warning-Ax-Katz-type estimates in many cases. Furthermore, our result can improve Cao-Sun's reduction recently obtained on counting the number of zeros of general diagonal equations over finite fields.     

More zeros of krawtchouk polynomials

Three theorems are given for the integral zeros of Krawtchouk polynomials. First, five new infinite families of integral zeros for the binary (q = 2) Krawtchouk polynomials are found. Next, a lower bound is given for the next integral zero for the degree four polynomial. Finally, three new infinite families inq are found for the degree three polynomials. The techniques used are from elementary number theory.