Notes on linear factor polynomial deflation in polynomial bases

Scalar polynomials as approximations to more general scalar functions lead to the study of scalar polynomials represented in a variety of classical systems of polynomials, including orthogonal systems and Lagrange polynomials, for example. This article, motivated in part by analogy with the existing methods for linear factor polynomial deflation in the monomial basis, finds forward and backward deflation formulae for several such representations. It also finds the sensitivity factor of the deflation process for each representation.     

关于对偶Steiner多项式的根的注记

受凸体的Steiner多项式的启发,定义了星体的对偶Steiner多项式,并利用对偶Aleksandrov-Fenchel不等式讨论了对偶Steiner多项式的根.进而,得到了关于对偶Steiner多项式的根的一些不等式,这些不等式恰好是关于Steiner多项式的根的不等式的对偶形式.     

Bounds on a polynomial

Methods are given for isolating and approximating the maxima, minima, and real roots of a polynomial with real coefficients. The methods are based on a variation diminishing property of the Bernstein coefficients of the polynomial and use of a recursive bisection technique.     

Pursuing polynomial bounds on torsion

It is well known that surface groups admit free and proper actions on finite products of infinite valence trees. In this note, we address the question of whether there can be a free and proper action on a finite product of bounded valence trees. We provide both some obstructions and an arithmetic criterion for existence. The bulk of the paper is devoted to an approach to verifying the arithmetic criterion which involves studying the character variety of certain surface groups over a field of positive characteristic. These methods may be useful for attempting to determine when groups admit good linear representations in other contexts.     

Remarks on a sine polynomial

Let n ≥ 0 be an integer. Then we have for ${x\in(0,\pi)}Let n ≥ 0 be an integer. Then we have for x ? (0,p){x\in(0,\pi)} :

?k=0n (( 2n+1) || (n-k ))\fracsin((2k+1)x)2k+1 ￡ \frac8n  n!(2n+1)!!.\sum_{k=0}^n { 2n+1 \choose n-k }\frac{\sin((2k+1)x)}{2k+1}\leq\frac{8^n \, n!}{(2n+1)!!}.      6. Estimates on polynomial exponential sums      J. Bourgain 《Israel Journal of Mathematics》2010,176(1):221-240 In this paper we establish new bounds on exponential sums of high degree for general composite moduli. The sums considered are either Gauss sums or ‘sparse’ and we rely on earlier work in the case of prime modulus.      7. Moment functions on polynomial hypergroups      Ágota Orosz  László Székelyhidi 《Archiv der Mathematik》2005,85(2):141-150 The study of moment functions on commutative topological groups leads to the study of systems of functional equations. Systems of functional equations characterizing moment functions and sequences of moment functions are closely related to exponential functions and additive functions. Here we describe moment functions and generalized moment functions on polynomial hypergroups.Received: 23 May 2003; revised: 20 September 2004      8. A note on polynomial cocycles      Robert Heaton 《Mathematische Zeitschrift》1961,76(1):235-239       9. Symmetrization operators on polynomial rings      A. Lascoux  M. P. Schützenberger 《Functional Analysis and Its Applications》1987,21(4):324-326       10. Variations on polynomial subgroup growth      Dan Segal 《Israel Journal of Mathematics》1996,94(1):7-19 A groupG hasweak polynomial subgroup growth (wPSG) of degree ≤α if each finite quotient Ḡ ofG contains at most │Ḡ│ a subgroups. The main result is that wPSG of degree α implies polynomial subgroup growth (PSG) of degree at mostf(α). It follows that wPSG is equivalent to PSG. A corollary is that if, in a profinite groupG, thek-generator subgroups have positive “density” δ, thenG is finitely generated (the number of generators being bounded by a function ofk and δ).      11. Nonnegative Morse polynomial functions and polynomial optimization      Lê Công-Trình 《Positivity》2016,20(2):483-498 In this paper we study the representation of Morse polynomial functions which are nonnegative on a compact basic closed semi-algebraic set in \(\mathbb R^{n}\), and having only finitely many zeros in this set. Following Bivià-Ausina (Math Z 257:745–767, 2007), we introduce two classes of non-degenerate polynomials for which the algebraic sets defined by them are compact. As a consequence, we study the representation of nonnegative Morse polynomials on these kinds of non-degenerate algebraic sets. Moreover, we apply these results to study the polynomial optimization problem for Morse polynomial functions.      12. From r-linearized polynomial equations to rm-linearized polynomial equations      《Finite Fields and Their Applications》2016       13. Spectral synthesis on multivariate polynomial hypergroups      László Székelyhidi 《Monatshefte für Mathematik》2008,153(2):145-152 In a recent paper [16] we presented some results concerning spectral analysis and spectral synthesis on polynomial hypergroups in a single variable. Now we show that using Hilbert’s Nullstellensatz, the Noether-Lasker-theorem and the Ehrenpreis-Palamodov-theorem the ideas of [16] can be extended to prove spectral analysis and spectral synthesis on any multivariate polynomial hypergroup. Author’s address: Institute of Mathematics, University of Debrecen, Egyetem tér 1, 4032 Debrecen, Hungary      14. Orthogonal polynomial expansions on sparse grids      Yanzhao Cao  Ying Jiang  Yuesheng Xu 《Journal of Complexity》2014       15. Imaginary part bounds on polynomial zeros      Lajos László 《Linear algebra and its applications》1982 Bounds for the imaginary parts of the zeros of a polynomial are given by the generalization of [6] and by the improvement of [3]. Methods of matrix theory are applied to orthogonal expansion of a polynomial.      16. A note on realizing polynomial algebras      J. Aguadé 《Israel Journal of Mathematics》1981,38(1-2):95-99 We characterize the polynomial algebras overZ which are realizable as the integral cohomology of some space, under the assumption that there are not two generators in the same dimension.      17. Self-commuting lattice polynomial functions on chains      Miguel Couceiro  Erkko Lehtonen 《Aequationes Mathematicae》2011,81(3):263-278 We provide sufficient conditions for a lattice polynomial function to be self-commuting. We explicitly describe self-commuting polynomial functions on chains.      18. Two remarks on the adjoint polynomial      Péter Csikvári 《European Journal of Combinatorics》2012,33(4):583-591       19. Computational bounds on polynomial differential equations   总被引：1，自引：0，他引：1   Daniel S. Graça  Jorge Buescu  Manuel L. Campagnolo 《Applied mathematics and computation》2009,215(4):1375-1385 In this paper we study from a computational perspective some properties of the solutions of polynomial ordinary differential equations.We consider elementary (in the sense of Analysis) discrete-time dynamical systems satisfying certain criteria of robustness. We show that those systems can be simulated with elementary and robust continuous-time dynamical systems which can be expanded into fully polynomial ordinary differential equations in Q[π]. This sets a computational lower bound on polynomial ODEs since the former class is large enough to include the dynamics of arbitrary Turing machines.We also apply the previous methods to show that the problem of determining whether the maximal interval of definition of an initial-value problem defined with polynomial ODEs is bounded or not is in general undecidable, even if the parameters of the system are computable and comparable and if the degree of the corresponding polynomial is at most 56.Combined with earlier results on the computability of solutions of polynomial ODEs, one can conclude that there is from a computational point of view a close connection between these systems and Turing machines.      20. Co-Poisson structures on polynomial Hopf algebras      Qi Lou  Quanshui Wu 《中国科学 数学(英文版)》2018,61(5):813-830 The Hopf dual H° of any Poisson Hopf algebra H is proved to be a co-Poisson Hopf algebra provided H is noetherian. Without noetherian assumption, unlike it is claimed in literature, the statement does not hold. It is proved that there is no nontrivial Poisson Hopf structure on the universal enveloping algebra of a non-abelian Lie algebra. So the polynomial Hopf algebra, viewed as the universal enveloping algebra of a finite-dimensional abelian Lie algebra, is considered. The Poisson Hopf structures on polynomial Hopf algebras are exactly linear Poisson structures. The co-Poisson structures on polynomial Hopf algebras are characterized. Some correspondences between co-Poisson and Poisson structures are also established.

Estimates on polynomial exponential sums

In this paper we establish new bounds on exponential sums of high degree for general composite moduli. The sums considered are either Gauss sums or ‘sparse’ and we rely on earlier work in the case of prime modulus.     

Moment functions on polynomial hypergroups

The study of moment functions on commutative topological groups leads to the study of systems of functional equations. Systems of functional equations characterizing moment functions and sequences of moment functions are closely related to exponential functions and additive functions. Here we describe moment functions and generalized moment functions on polynomial hypergroups.Received: 23 May 2003; revised: 20 September 2004     

Variations on polynomial subgroup growth

A groupG hasweak polynomial subgroup growth (wPSG) of degree ≤α if each finite quotient Ḡ ofG contains at most │Ḡ│ ^a subgroups. The main result is that wPSG of degree α implies polynomial subgroup growth (PSG) of degree at mostf(α). It follows that wPSG is equivalent to PSG. A corollary is that if, in a profinite groupG, thek-generator subgroups have positive “density” δ, thenG is finitely generated (the number of generators being bounded by a function ofk and δ).     

Nonnegative Morse polynomial functions and polynomial optimization

In this paper we study the representation of Morse polynomial functions which are nonnegative on a compact basic closed semi-algebraic set in \(\mathbb R^{n}\), and having only finitely many zeros in this set. Following Bivià-Ausina (Math Z 257:745–767, 2007), we introduce two classes of non-degenerate polynomials for which the algebraic sets defined by them are compact. As a consequence, we study the representation of nonnegative Morse polynomials on these kinds of non-degenerate algebraic sets. Moreover, we apply these results to study the polynomial optimization problem for Morse polynomial functions.     

Spectral synthesis on multivariate polynomial hypergroups

In a recent paper [16] we presented some results concerning spectral analysis and spectral synthesis on polynomial hypergroups in a single variable. Now we show that using Hilbert’s Nullstellensatz, the Noether-Lasker-theorem and the Ehrenpreis-Palamodov-theorem the ideas of [16] can be extended to prove spectral analysis and spectral synthesis on any multivariate polynomial hypergroup. Author’s address: Institute of Mathematics, University of Debrecen, Egyetem tér 1, 4032 Debrecen, Hungary     

Imaginary part bounds on polynomial zeros

Bounds for the imaginary parts of the zeros of a polynomial are given by the generalization of [6] and by the improvement of [3]. Methods of matrix theory are applied to orthogonal expansion of a polynomial.     

A note on realizing polynomial algebras

We characterize the polynomial algebras overZ which are realizable as the integral cohomology of some space, under the assumption that there are not two generators in the same dimension.     

Self-commuting lattice polynomial functions on chains

We provide sufficient conditions for a lattice polynomial function to be self-commuting. We explicitly describe self-commuting polynomial functions on chains.     

Computational bounds on polynomial differential equations

In this paper we study from a computational perspective some properties of the solutions of polynomial ordinary differential equations.We consider elementary (in the sense of Analysis) discrete-time dynamical systems satisfying certain criteria of robustness. We show that those systems can be simulated with elementary and robust continuous-time dynamical systems which can be expanded into fully polynomial ordinary differential equations in Q[π]. This sets a computational lower bound on polynomial ODEs since the former class is large enough to include the dynamics of arbitrary Turing machines.We also apply the previous methods to show that the problem of determining whether the maximal interval of definition of an initial-value problem defined with polynomial ODEs is bounded or not is in general undecidable, even if the parameters of the system are computable and comparable and if the degree of the corresponding polynomial is at most 56.Combined with earlier results on the computability of solutions of polynomial ODEs, one can conclude that there is from a computational point of view a close connection between these systems and Turing machines.     

Co-Poisson structures on polynomial Hopf algebras

The Hopf dual H° of any Poisson Hopf algebra H is proved to be a co-Poisson Hopf algebra provided H is noetherian. Without noetherian assumption, unlike it is claimed in literature, the statement does not hold. It is proved that there is no nontrivial Poisson Hopf structure on the universal enveloping algebra of a non-abelian Lie algebra. So the polynomial Hopf algebra, viewed as the universal enveloping algebra of a finite-dimensional abelian Lie algebra, is considered. The Poisson Hopf structures on polynomial Hopf algebras are exactly linear Poisson structures. The co-Poisson structures on polynomial Hopf algebras are characterized. Some correspondences between co-Poisson and Poisson structures are also established.