Abstract: We present a new approach to constructing pseudorandom generators that fool low-degree polynomials over finite fields, based on the Gowers norm. Using this approach, we obtain the following main constructions of explicitly computable generators G : FsrarrFn that fool polynomials over a prime field F: (1) a generator that fools degree-2 (i.e., quadratic) polynomials to within error 1/n, with seed length s = O(log n); (2) a generator that fools degree-3 (i.e., cubic) polynomials to within error epsiv, with seed length s = O(Iog|F| n) + f(epsiv, F) where f depends only on epsiv and F (not on n), (3) assuming the "Gowers inverse conjecture," for every d a generator that fools degree-d polynomials to within error epsiv, with seed length, s = O(dldrIog|F| n) + f(d, epsiv, F) where f depends only on d, epsiv, and F (not on n). We stress that the results in (1) and (2) are unconditional, i.e. do not rely on any unproven assumption. Moreover, the results in (3) rely on a special case of the conjecture which may be easier to prove. Our generator for degree-d polynomials is the component-wise sum of d generators for degree-l polynomials (on independent seeds). Prior to our work, generators with logarithmic seed length were only known for degree-1 (i.e., linear) polynomials (Naor and Naor; SIAM J. Comput., 1993). In fact, over small fields such as F2 = {0,1}, our results constitute the first progress on these problems since the long-standing generator by Luby, Velickovic and Wigderson (ISTCS1993), whose seed length is much bigger: s = exp (Omega(radiclogn)), even for the case of degree-2 polynomials over F2.

Abstract: Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. In recent papers, we derived closed-form polynomials that are orthonormal over a hexagonal pupil, such as the hexagonal segments of a large mirror. We extend our work to elliptical, rectangular, and square pupils. Using the circle polynomials as the basis functions for their orthogonalization over such pupils, we derive closed-form polynomials that are orthonormal over them. These polynomials are unique in that they are not only orthogonal across such pupils, but also represent balanced classical aberrations, just as the Zernike circle polynomials are unique in these respects for circular pupils. The polynomials are given in terms of the circle polynomials as well as in polar and Cartesian coordinates. Relationships between the orthonormal coefficients and the corresponding Zernike coefficients for a given pupil are also obtained. The orthonormal polynomials for a one-dimensional slit pupil are obtained as a limiting case of a rectangular pupil.

Abstract: In this paper we present applications of a special class of homogeneous monogenic polynomials constructed, in the framework of hypercomplex function theory, in order to be an Appell set of polynomials. In particular, we derive important properties of an associated exponential function from R3 to R3 and propose a generalization to Rn+1.

Abstract: Toom-Cook strategy is a well-known method for building algorithms to efficiently multiply dense univariate polynomials. Efficiency of the algorithm depends on the choice of interpolation points and on the exact sequence of operations for evaluation and interpolation. If carefully tuned, it gives the fastest algorithm for a wide range of inputs. This work smoothly extends the Toom strategy to polynomial rings, with a focus on . Moreover a method is proposed to find the faster Toom multiplication algorithm for any given splitting order. New results found with it, for polynomials in characteristic 2, are presented. A new extension for multivariate polynomials is also introduced; through a new definition of density leading Toom strategy to be efficient.

Abstract: Employing the random matrix formulation of Chern-Simons theory on Seifert manifolds, we show how the Stieltjes-Wigert orthogonal polynomials are useful in exact computations in Chern-Simons matrix models. We construct a biorthogonal extension of the Stieltjes-Wigert polynomials, not available in the literature, necessary to study Chern-Simons matrix models when the geometry is a lens space. We also study the relationship between Stieltjes-Wigert and Rogers-Szego polynomials and the corresponding equivalence with a unitary matrix model. Finally, we give a detailed proof of a result that relates quantum dimensions with averages of Schur polynomials in the Stieltjes-Wigert ensemble.

Abstract: In the recent paper the interesting q-Euler numbers and polynomials introduced in JMAA. The purpose of this paper is to construct the modified q-Euler numbers and polynomiasl. Finally we will give the interesting many identities related to these numbers and polynomials.

Abstract: We briefly review the five possible real polynomial solutions of hypergeometric differential equations. Three of them are the well known classical orthogonal polynomials, but the other two are different with respect to their orthogonality properties. We then focus on the family of polynomials which exhibits a finite orthogonality. This family, to be referred to as the Romanovski polynomials, is required in exact solutions of several physics problems ranging from quantum mechanics and quark physics to random matrix theory. It appears timely to draw attention to it by the present study. Our survey also includes several new observations on the orthogonality properties of the Romanovski polynomials and new developments from their Rodrigues formula.

Abstract: This note investigates the gap existing between positive polynomials and sum of squares (SOS) of polynomials, which affects several analysis and synthesis tools in control systems based on polynomial SOS relaxations, and about which almost nothing is known. In particular, a matrix characterization of the PNS, that is the positive homogeneous forms that are not SOS, is proposed, which allows to show that any PNS is the vertex of an unbounded cone of PNS. Moreover, a complete parametrization of the set of PNS is introduced.

Abstract: A general theoretical approach has been developed for the determination of orthonormal polynomials over any integrable domain, such as a hexagon. This approach is better than the classical Gram-Schmidt orthogonalization process because it is nonrecursvie and can be performed rapidly with matrix transformations. The determination of the orthonormal hexagonal polynomials is demonstrated as an example of the matrix approach.

Abstract: In 1888, Hilbert described how to find real polynomials in more than one variable which take only non-negative values but are not a sum of squares of polynomials. His construction was so restrictive that no explicit examples appeared until the late 1960s. We revisit and generalize Hilbert's construction and present many such polynomials.

Abstract: In this paper we study q-Euler numbers and polynomials by using p-adic q-fermionic integrals on Z_p. The methods to study q-Euler numbers and polynomials in this paper are new.

Abstract: ix List of Symbols Used x

Abstract: Zeros of orthogonal polynomials have had a fascination at least since Gauss’ discovery that optimal quadrature for the Ftiemann integral on [-I, 11 involves the zeros of the Legendre polynomials. A special reason for recent interest concerns the fact that zeros are eigenvalues of cutoff finite difference matrices. Explicitly, if P,, p , are the monic orthogonal and orthonormal polynomials for OPRL (RL = real line) and a,, pn for OPUC (UC = unit circle), then

Abstract: In this article, we present new bounds for the zeros of polynomials depending on some estimates for the spectral norms and the spectral radii of the square and the cube of the Frobenius companion matrix.

Abstract: We consider real polynomials in finitely many variables. Let the variables consist of finitely many blocks that are allowed to overlap in a certain way. Let the solution set of a finite system of polynomial inequalities be given, where each inequality involves only variables of one block. We investigate polynomials that are positive on such a set and sparse in the sense that each monomial involves only variables of one block. In particular, we derive a short and direct proof for Lasserre’s theorem on the existence of sums of squares certificates respecting the block structure. The motivation for the results can be found in the literature on numerical methods for global optimization of polynomials that exploit sparsity.

Abstract: We establish a best possible extension of a famous Theorem of Vietoris about the positivity of a general class of cosine sums. Our result refines and sharpens several earlier generalizations of this Theorem, and settles some open questions regarding the possibility of further improvement of it. Some new inequalities for trigonometric sums are given. We show that our results have applications within the context of positive sums of Gegenbauer polynomials and quadrature methods. We also obtain some existing estimates for the location of zeros of certain trigonometric polynomials under a weakened condition on their coefficients.

Abstract: We propose the idea of building a secure hash using quadratic or higher degree multivariate polynomials over a finite field as the compression function. We analyze some security properties and potential feasibility, where the compression functions are randomly chosen high-degree polynomials, and show that under some plausible assumptions, high-degree polynomials as compression functions has good properties. Next, we propose to improve on the efficiency of the system by using some specially designed polynomials generated by a small number of random parameters, where the security of the system would then relies on stronger assumptions, and we give empirical evidence for the validity of using such polynomials.

Abstract: In the recent progress (4), (17) and (25), the well-known JC (Ja- cobian conjecture) ((2), (10)) has been reduced to a VC (vanishing conjecture) on the Laplace operators and HN (Hessian nilpotent) polynomials (the poly- nomials whose Hessian matrix are nilpotent). In this paper, we first show that the vanishing conjecture above, hence also the JC, is equivalent to a van- ishing conjecture for all 2nd order homogeneous dierential operators and -nilpotent polynomials P (the polynomials P(z) satisfying m P m = 0 for all m > 1). We then transform some results in the literature on the JC, HN poly- nomials and the VC of the Laplace operators to certain results on -nilpotent polynomials and the associated VC for 2nd order homogeneous dierential operators . This part of the paper can also be read as a short survey on HN polynomials and the associated VC in the more general setting. Finally, we discuss a still-to-be-understood connection of -nilpotent polynomials in gen- eral with the classical orthogonal polynomials in one or more variables. This connection provides a conceptual understanding for the isotropic properties of homogeneous -nilpotent polynomials for 2nd order homogeneous full rank dierential operators with constant coecients.

Abstract: In this paper, we characterize all the Darboux polynomials of the Chen system, , and prove that the system is not algebraic integrability. For proving the results, we use the weight homogeneous polynomials and the method of characteristics.

Abstract: Using non-archimedean q-integrals on Z"p defined in [T Kim, On a q-analogue of the p-adic log gamma functions and related integrals, J Number Theory 76 (1999) 320-329; T Kim, q-Volkenborn integration, Russ J Math Phys 9 (2002) 288-299], we define new Changhee q-Euler polynomials and numbers which are different from those of Kim [T Kim, p-adic q-integrals associated with the Changhee-Barnes' q-Bernoulli polynomials, Integral Transforms Spec Funct 15 (2004) 415-420] and Carlitz [L Carlitz, q-Bernoulli and Eulerian numbers, Trans Amer Math Soc 76 (1954) 332-350] We define generating functions of multiple q-Euler numbers and polynomials Furthermore we construct a multivariate Hurwitz type zeta function which interpolates the multivariate q-Euler numbers or polynomials at negative integers

Abstract: We give, over a finite field F q , explicit factorizations into a product of irreducible polynomials, of the cyclotomic polynomials of order 3·2n, the Dickson polynomials of the first kind of order 3·2nand the Dickson polynomials of the second kind of order 3·2ni¾? 1.