Hilbert's Tenth Problem for fields of rational functions over finite fields

Summary We prove that there is no algorithm to solve arbitrary polynomial equations over a field of rational functions in one letter with constants in a finite field of characteristic other than 2 and hence, Hilbert's Tenth Problem for any such field is undecidable.Oblatum 1-XI-1989Supported in part by NSF Grant DMS 8605198.     

Degenerate difference-differential equations: Algebraic theory

In this paper we consider the algebraic aspects of the theory of degenerate difference-differential equations. It will be shown that the fundamental algebraic concepts to be used are module theoretic. We have to consider similarity of polynomial matrices in one or more indeterminates. In the case of systems with commensurable lags the underlying modules have a simple structure, because the corresponding ring of scalars is the principal ideal domain of polynomials in one indeterminate. This fact makes it possible to prove a structure theorem for degenerate difference-differential equations with commensurable lags. This theorem shows that degenerate systems of this type essentially are trivial in the sense of Henry [15], i.e., the characteristic quasipolynomial is a polynomial. Further it is shown that coordinate transforms with “time lag” play an essential role for the construction of degenerate equations. The power of the method is demonstrated by some examples, some of which are equations with incommensurable lags.     

双比例延迟微分方程配置法的可达阶

近几十年来,随着延迟微分方程广泛应用于变最测试、信号传递、机械化工、生命科学及经济管理系统等实际中,人们对此类方程的数值计算要求越来越高．     

DARBOUX POLYNOMIALS AND NON-ALGEBRAIC INTEGRABILITY OF THE L SYSTEM

In this paper, we characterize all of the Darboux polynomials of the L system and prove that the system is not algebraically integrable, using the weight homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations.     

On the best approximation in the mean of continuous functions by polynomials,which are majorized by a given function

We discuss the problem of the best approximation in the mean of functions, which are continuous on a segment, by the polynomials in a differentiable Chebyshev system that do not exceed a given differentiable function. We prove that the extremal polynomial is unique and develop its characteristic properties.Translated from Matematicheskie Zametki, Vol. 11, No. 4, pp. 365–374, April, 1972.The author thanks A. L. Garkavi for posing the problem.     

Some Results on Ordinary Differential Operators with Periodic Coefficients

For a general ordinary differential operator \(\mathcal {L}\) with periodic coefficients we prove that the characteristic polynomial of the Floquet matrix is irreducible over the field of meromorphic functions. We also consider a multipoint eigenvalue problem and show that its eigenspaces are spanned by pure or generalized Floquet solutions. Finally, at the end of the paper we mention some relevant conjectures and open questions.     

Conditions for polynomial Liénard centers

In 1999, Christopher gave a necessary and sufficient condition for polynomial Li′enard centers, which requires coupled functional equations, where the primitive functions of the damping function and the restoring function are involved, to have polynomial solutions. In order to judge whether the coupled functional equations are solvable, in this paper we give an algorithm to compute a Gr¨obner basis for irreducible decomposition of algebraic varieties so as to find algebraic relations among coefficients of the damping function and the restoring function. We demonstrate the algorithm for polynomial Li′enard systems of degree 5, which are divided into 25 cases. We find all conditions of those coefficients for the polynomial Li′enard center in 13 cases and prove that the origin is not a center in the other 12 cases.     

Additive schwarz methods for indefinite hypersingular integral equations in R3 - the p-version

We propose and analyze preconditioners for the p-version of the boundary element method in three dimensions. We consider indefinite hypersingular integral equations on surfaces and use quadrilateral elements for the boundary discretization. We use the GMRES method as iterative solver for the linear systems and prove for an overlapping additive Schwarz method that the number of iterations is bounded. This bound is independent of the polynomial degree of the ansatz functions and of the size of the underlying mesh. For a modified diagonal scaling, which uses special basis functions, we prove that the number of iterations grows only polylogarithmically in the polynomial degree. Here, a sufficiently fine mesh is required. Numerical results supporting the theory are presented.     

Fast Linear Homotopy to Find Approximate Zeros of Polynomial Systems

We prove a new complexity bound, polynomial on the average, for the problem of finding an approximate zero of systems of polynomial equations. The average number of Newton steps required by this method is almost linear in the size of the input (dense encoding). We show that the method can also be used to approximate several or all the solutions of non-degenerate systems, and prove that this last task can be done in running time which is linear in the Bézout number of the system and polynomial in the size of the input, on the average.     

Linear relations among algebraic solutions of differential equations

For any univariate polynomial with coefficients in a differential field of characteristic zero and any integer, q, there exists an associated nonzero linear ordinary differential equation (LODE) with the following two properties. Each term of the LODE lies in the differential field generated by the rational numbers and the coefficients of the polynomial, and the qth power of each root of the polynomial is a solution of this LODE. This LODE is called a qth power resolvent of the polynomial. We will show how one can get a resolvent for the logarithmic derivative of the roots of a polynomial from the αth power resolvent of the polynomial, where α is an indeterminate that takes the place of q. We will demonstrate some simple relations among the algebraic and differential equations for the roots and their logarithmic derivatives. We will also prove several theorems regarding linear relations of roots of a polynomial over constants or the coefficient field of the polynomial depending upon the (nondifferential) Galois group. Finally, we will use a differential resolvent to solve the Riccati equation.     

Projection Volumes of Hyperplane Arrangements

We prove that for any finite real hyperplane arrangement the average projection volumes of the maximal cones are given by the coefficients of the characteristic polynomial of the arrangement. This settles the conjecture of Drton and Klivans that this held for all finite real reflection arrangements. The methods used are geometric and combinatorial. As a consequence, we determine that the angle sums of a zonotope are given by the characteristic polynomial of the order dual of the intersection lattice of the arrangement.     

Cauchy problem for a class of integro-differential equations

We consider the Cauchy problem for integro-differential equations containing Wiener-Hopf operators. We define the characteristic polynomial of the problem. We show that if all roots of the characteristic polynomial have negative imaginary parts and the corresponding strong moment problem is solvable, then the problem is equivalent to a Wiener-Hopf integral equation. We consider an example of an application of the result to a problem related to nonlocal wave interaction.     

On Smale's 17th Problem: A Probabilistic Positive Solution

Smale's 17th Problem asks “Can a zero of n complex polynomial equations in n unknowns be found approximately, on the average [for a suitable probability measure on the space of inputs], in polynomial time with a uniform algorithm?” We present a uniform probabilistic algorithm for this problem and prove that its complexity is polynomial. We thus obtain a partial positive solution to Smale's 17th Problem.     

On rational integrability of Euler equations on Lie algebra so(4, ℂ)

We consider the Euler equations on the Lie algebra so(4, ℂ) with a diagonal quadratic Hamiltonian. It is known that this system always admits three functionally independent polynomial first integrals. We prove that if the system has a rational first integral functionally independent of the known three ones so called fourth integral, then it has a polynomial first integral that is also functionally independent of them. This is a consequence of more general fact that for these systems the existence of Darboux polynomial with no vanishing cofactor implies the existence of polynomial fourth integral.     

Study of the Convergence of Interpolation Processes with Splines of Even Degree

Siberian Mathematical Journal - We study the convergence of interpolation processes by Subbotin polynomial splines of even degree. We prove that the good conditionality of a system of equations for...     

关于格的卡氏积的特征多项式和几何性

利用格L_i(i=1,2)的性质研究了它们的卡氏积L=L_1×L_2的性质,得到了L的秩函数、Mbius函数和特征多项式,并且由L_i的几何性证明了L的几何性.     

A Note on Meromorphic Solutions of Linear Partial Differential Equations of Second Order

We will prove a uniqueness theorem for meromorphic solutions of linear partial differential equations of second order with polynomial coefficients associated with the Jacobi differential equations and their generalizations.     

The characteristic polynomial of a multiarrangement

Given a multiarrangement of hyperplanes we define a series by sums of the Hilbert series of the derivation modules of the multiarrangement. This series turns out to be a polynomial. Using this polynomial we define the characteristic polynomial of a multiarrangement which generalizes the characteristic polynomial of an arrangement. The characteristic polynomial of an arrangement is a combinatorial invariant, but this generalized characteristic polynomial is not. However, when the multiarrangement is free, we are able to prove the factorization theorem for the characteristic polynomial. The main result is a formula that relates ‘global’ data to ‘local’ data of a multiarrangement given by the coefficients of the respective characteristic polynomials. This result gives a new necessary condition for a multiarrangement to be free. Consequently it provides a simple method to show that a given multiarrangement is not free.     

线性变系数中立型变延迟微分方程谱方法的收敛性

本文主要研究了应用谱方法求解线性变系数中立型变延迟微分方程,构造了相应的基于Chebyshev和Legendre正交多项式的数值方法, 证明了其收敛性,最后给出了数值算例. 这些结果表明应用谱方法求解延迟微分方程可以获得谱收敛与谱精度的计算效果.     

Sur la lissification de type Płoski–Popescu

We prove a version of the Popescu's smoothing theorem for W-systems defined by J. Denef and L. Lipschitz. This generalizes P?oski's version for analytic equations in characteristic zero.