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Topic: Laurent polynomials

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	ipedia.com: Knot polynomial Article (Site not responding. Last check: 2007-10-29)
		Technically, an Alexander polynomial is a generator of a principal Alexander ideal related to the homology of the infinitely cyclic cover of a knot complement—where all the emphasised phrases have particular mathematical meanings.
		(this is an Alexander polynomial of the knot).
		HOMFLYPT is a binary (two-variable) polynomial, with as with the predecessors.
	www.ipedia.com /knot_polynomial.html (1125 words)

	math lessons - Laurent series
		In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree.
		Laurent series with complex coefficients are an important tool in complex analysis, especially to investigate the behavior of functions near singularities.
		Two such formal Laurent series may be added by adding the coefficients, and because of the finiteness of the negative-degree coefficients, they may also be multiplied using convolution of the coefficient sequences.
	www.mathdaily.com /lessons/Laurent_polynomial (946 words)

	[No title]
		The structure of these Laurent polynomials was what led Mills, Robbins and Rumsey to define alternating sign matrices in the first place, and it turns out that a very perspicuous way of understanding the terms of these Laurent polynomials is to view them as domino tilings of Aztec diamonds in algebraic clothing.
		I've conjectured that in this Laurent polynomial, every coefficient is equal to 1, and that moreover the exponents of the x-variables and z-variables are always +1, 0 or -1 while the exponents of the y-variables are always -1, 0, 1, 2, 3, or 4.
		For each monomial in one of the Laurent polynomials hex(i,j,k), the exponents of the x-variables naturally form a triangular array of integers (the entries of which conjecturally must be in the set {-1,0,+1}).
	www.math.wisc.edu /~propp/cube-recur (1617 words)

	Shift operator - Wikipedia, the free encyclopedia
		Spaces of polynomials carry numerous topological structures; shift operators can be constructed by extension on corresponding complete spaces.
		One can say that the analogue in this case of the polynomial representation is that by Laurent polynomials.
		The theory of analytic functions is related to that of polynomials, by allowing infinite power series; on the other hand meromorphic functions have Laurent series that terminate in the direction of negative exponents.
	en.wikipedia.org /wiki/Shift_operator (379 words)

	[ref] 60 Polynomials and Rational Functions
		A Laurent polynomial is a univariate rational function whose denominator is a monomial.
		Laurent polynomials also can be considered as quotients of a univariate polynomial by a power of the indeterminate.
		The addition and multiplication of univariate polynomials extends to Laurent polynomials (though it might be impossible to interpret a Laurent polynomial as a function) and many functions for univariate polynomials extend to laurent polynomials (or extended versions for laurent polynomials exist).
	www.karlin.mff.cuni.cz /asc/network/prirucky/gap/ref/CHAP060.htm (3306 words)

	Monotonicity Of Zeros Of Orthogonal Laurent Polynomials (ResearchIndex)
		Abstract: Monotonicity of zeros of orthogonal Laurent polynomials associated with a strong distribution with respect to a parameter is discussed.
		6 Orthogonal Laurent polynomials and Gaussian quadrature (context) - Jones, Thron - 1981
		4 Orthogonal Laurent polynomials (context) - Hendriksen, van Rossum - 1986
	citeseer.ist.psu.edu /498800.html (392 words)

	1. Introduction
		He used polynomials as rules by giving an admissible term ordering on the terms and using the largest monomial according to this ordering as a left hand side of a rule.
		A Gröbner basis now is a set of polynomials G such that every polynomial in the polynomial ring has a unique normal form with respect to reduction using the polynomials in G as rules (especially the polynomials in the ideal generated by G reduce to zero using G).
		For the polynomial ring Buchberger developed a terminating procedure to transform a finite generating set of a polynomial ideal into a finite Gröbner basis of the same ideal.
	www.mathematik.uni-kl.de /~zca/Reports_on_ca/09/paper_html/node1.html (747 words)

	GAP Manual: 19 Polynomials (Site not responding. Last check: 2007-10-29)
		The set of all Laurent polynomials L(R) over a ring R together with above definitions of + and * is again a ring, the Laurent polynomial ring over R, and R is called the base ring of L(R).
		As was already noted in the introduction to this chapter polynomial rings are rings, so all ring functions (see chapter Rings) are applicable to polynomial rings.
		As was already noted in the introduction to this chapter Laurent polynomial rings are rings, so all ring functions (see chapter Rings) are applicable to polynomial rings.
	www.math.jussieu.fr /~jmichel/htm/CHAP019.htm (2791 words)

	Laurent Series
		Laurent series for the inversion of perturbed linear operators on Hilbert space.
		The connection between the extremality of the coefficients of a Laurent series and the completeness of a set of analytic functions in the space of continuous functions.
		A certain analogue of the Laurent series for functions of several complex variables that are holomorphic on strongly linearly convex sets.
	math.fullerton.edu /mathews/c2003/LaurentSeriesBib/Links/LaurentSeriesBib_lnk_3.html (795 words)

	REACH: A Dozen Laurent Recurrences
		In each case other than #11, the intent is that you should type f(n) (or f(n,0) or f(n,0,0), as needed) for various small values of n and observe that the output in each case is a Laurent polynomial; that is, when it is written as a rational function, the denominator is just a product.
		These polynomials have an even clearer relationship to the combinatorial model of matchings-of-the-Aztec-diamond-graph than the Laurent polynomials of the preceding example.
		All the coefficients in the Laurent polynomial f(n,k) are equal to 1, and the individual monomials correspond to the perfect matchings of certain graphs (whose structure became clear through email conversations between Ron Adin, Robin Chapman, Ira Gessel, Rick Kenyon, Christian Krattenthaler, Eric Kuo, and myself).
	www.math.harvard.edu /~propp/reach/laurent.html (1445 words)

	Publications of the SPACES team
		It was shown in a previous work that this ability to change posture without meeting a singularity is equivalent to the existence of a point in the workspace, such that a polynomial of degree four depending on the parameters of the manipulator and on the cartesian coordinates of the effector has a triple root.
		Let f1,andhellip;,fk be k multivariate polynomials which have a finite number of common zeros in the algebraic closure of the ground field, counting the common zeros at infinity.
		When all the polynomials have the same degree, the complexity of this algorithm is polynomial relative to the generic number of solutions.
	www-calfor.lip6.fr /~safey/Spaces/publications.html (13078 words)

	Página profesional de Andrei
		Martínez-Finkelshtein, Bernstein-Szegö's theorem for Sobolev orthogonal polynomials, Constr.
		Entropy-like functionals for Orthogonal Polynomials, Asymptotics and some Physical Applications, Invited talk (joint with J. Sánchez Dehesa) at V International Symposium on Orthogonal Polynomials, Special Functions and Their Applications, Patras, Greece, September 20-24, 1999.
		On the electrostatic interpretation of the zeros of Jacobi polynomials, Contributed talk at INTAS Workshop "Rational Approximation and its Applications", Coimbra, Portugal, February 7 - 9, 2002.
	www.ual.es /personal/andrei/publ-en.html (1411 words)

	[No title]
		But wait, in the ring of Laurent polynomials, not every nonzero element is invertible.
		You know the routine for computing the GCD of the first two polynomials: subtract and so replace the second by x+1, then subtract a multiple of this from the first to get 0; arriving at 0 informs us that the last term (x+1) is the GCD.
		('degree' for Laurent polynomials is the difference between the greatest and least exponents used).
	www.math.niu.edu /~rusin/known-math/97/smith.nf (1000 words)

	CPC Licence Alert (Site not responding. Last check: 2007-10-29)
		Symbolic computer algebra is used to automate the Euclidean algorithm for Laurent polynomials [1] so as to factorise wavelet transforms yielding a sequence of simple arithmetic operations suitable for parallel, in-place, implementation [2].
		The program requires that the Laurent polynomial quotients used in the algorithm have Laurent degree at most 1.
		Using Grbner bases and polynomial reduction on the filter coefficients reduces the number of unknown coefficients appearing in expressions.
	www.cpc.cs.qub.ac.uk /summaries/ADLE.html (188 words)

	[No title]
		A Laurent polynomial is a linear combination of monomials that may have negative integer exponents as well as positive integer exponents.
		For example $x+1+x sup -1 $ is a Laurent polynomial in one variable and $x+y+ x sup -1 y sup 2 $ is one in two variables.
		A Laurent polynomial G is \fIanti-symmetric\fP if for any w in the Weyl group W, w(G)=sgn(w)G. stands for "the constant term of" (in $x= (x sub 1,...
	www.math.temple.edu /~zeilberg/TROFF/unifiedmac.troff (5139 words)

	Monique Laurent
		Three lectures on Moment Matrices and Optimization over Polynomials as part of the ADONET Doctoral school on Optimization over Polynomials and Semidefinite Programming organized at the University of Klagenfurt, September 12-16, 2005.
		Polynomial optimization and sums of squares of polynomials.
		Polynomial instances of the positive semidefinite and euclidean distance matrix completion problems.
	homepages.cwi.nl /~monique (839 words)

	LP class
		This class represents Laurent polynomials with terms of integer (positive or negative) powers and rational coefficients.
		The unit polynomial is represented as a rational scalar with its numerator equal to its denominator.
		The zero polynomial is represented as a rational scalar with a zero numerator.
	www.damtp.cam.ac.uk /user/na/people/Malcolm/primus/ex2/LP.htm (572 words)

	[No title]
		Most intriguingly (for me), if we record, for each polynomial, the coefficient of the highest-degree monomial, we get the integer sequence 1 2 2 6 16 16 176 512 512 22016 65536 65536 Two thirds of these terms are powers of two.
		In situations where the original recurrence gives rise to Laurent polynomials in which all coefficients are 1, you won't get anything really new out of this (assuming that Laurentness holds at all).
		E.g., there are cases in which the Laurent polynomial x should be viewed as 1/(x^{-1}).
	www.math.harvard.edu /~propp/reach/Projects-2003 (3198 words)

	Fermat User's Guide
		The highest precedence polynomial variable in x is replaced with the matrix [y] and the resulting expression simplified.
		If there are several polynomial variables, the exact coefficient desired is specified by listing the exponents of the variables in precedence order, highest first.
		For many matrices of polynomials, especially over the ground ring of integers or rationals, it is faster to figure out the degree of the determinant, evaluate the determinant at a set of values (say, integers), and then interpolate to compute the determinant from these values.
	www.bway.net /~lewis/fermat/ferman.html (16650 words)

	MIT Combinatorics Seminar: Eric Rains (Site not responding. Last check: 2007-10-29)
		I'll discuss two families of Laurent polynomials with hyperoctahedral symmetry, both indexed by partitions.
		The first, Koornwinder's orthogonal polynomials, includes all of the classical (and quantum classical) spherical functions and characters as special and limiting cases, as well as the Jack and Macdonald polyomials.
		The second, Okounkov's interpolation polynomials, is defined (and overdetermined) by specifying a large collection of zeros.
	www-math.mit.edu /~combin/archive/2004_fall/04_11_24_rains.html (110 words)

	[No title]
		Inequalities for zeros of associated polynomials and derivatives of orthogonal polynomials, Appl.
		Chebyshev-Laurent polynomials and weighted approximation in "Orthogonal Functions, Moment Theory, and Continued Fractions: Theory and Applications" (W. Jones and A. Sri Ranga eds.), Lecture Notes in Pure and Applied Mathematics 199, Marcel Dekker, 1998, pp.
		On a conjecture concerning monotonicity of zeros of ultraspherical polynomials, J. Approx.
	www.dcce.ibilce.unesp.br /~dimitrov/bib.htm (603 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		% wave2lp - Laurent polynomial associated to a wavelet.
		% horzcat - Horizontal concatenation of Laurent polynomials.
		% lp2num - Coefficients of a Laurent polynomial object.
	www.clemson.edu /cle4_share/CWE/COES0915_CLUG/REFERENCE/matlabr14/toolbox/wavelet/wavelet/lifting.m (1273 words)

	The Fast Lifting Wavelet Transform
		This summation is also known as a Laurent polynomial or Laurent series[3].
		Then we lift one of these streams as in the left of figures 4 and 5 by applying a Laurent polynomial to the other and adding it to the first.
		In other words, inverting a lifting transform is the same as changing all the signs of the lifting Laurent polynomials in figures 4 and 5 and run it backwards, i.e.
	perso.wanadoo.fr /polyvalens/clemens/lifting/lifting.html (5432 words)

	Walter Van Assche
		A continuum limit of the relativistic Toda lattice: asymptotic theory of discrete Laurent orthogonal polynomials with varying recurrence coefficients (with J. Coussement), J. Phys.
		Perturbation of orthogonal polynomials on an arc of the unit circle, II, (with L. Golinskii, P. Nevai and F. Pinter), J. Approx.
		Perturbation of orthogonal polynomials on an arc of the unit circle (with L. Golinskii and P. Nevai), J. Approx.
	wis.kuleuven.be /analyse/walter (1542 words)

	Systems of Laurent Polynomial Equations and Gröbner Bases (Site not responding. Last check: 2007-10-29)
		But the degrees of these polynomials could be very high, hence in this talk I shall propose another method to solve Laurent polynomial equations.
		This method is based on results of the paper PU (Pauer, F., Unterkircher, A.: Gröbner Bases for Ideals in Laurent polynomial rings and their Application to Systems of Difference Equations.
		Let J be the ideal in R generated by the Laurent polynomials
	www.math.unm.edu /ACA/1998/sessions/gb/pauer (276 words)

	GAP Manual: 19 Polynomials
		A (univariate) Laurent polynomial over R is a sequence (..., c_{-1}, c_0, c_1,...) of elements of R such that only finitely many are non-zero.
		First of all, the polynomial (..., x_0 = 0, x_1 = 1, x_2 = 0,...) is commonly denoted by x and is called an indeterminate over R, (..., c_{-1}, c_0, c_1,...) is written as...
		The next sections describe the ring functions applicable to Laurent Ring Functions for Laurent Polynomial Rings).
	www-gap.dcs.st-and.ac.uk /oldsite/Manual/C019S000.htm (720 words)

	GAP Manual: 63. Weyl Groups and Hecke Algebras
		In this chapter we describe functions for dealing with finite Weyl groups, associated Hecke algebras over a ring of Laurent polynomials, and their representations.
		Usually, A is chosen to be the ring of Laurent polynomials in an indeterminate v, and all q_i are chosen to be v^2.
		We wish to work over the ring of Laurent polynomials in an indeterminate v, and we want to have u=v^2 as a parameter for the generic Hecke algebra H of W.
	www.math.uiuc.edu /Software/GAP-Manual/Weyl_Groups_and_Hecke_Algebras.html (1544 words)

	The Wavelet Digest :: View topic - Preprint: Spectral factorization of Laurent polynomials
		factoring a Laurent polynomial, which is positive on the unit circle,
		of the coefficients of the Laurent polynomial, by the closeness of the
		zeros of this polynomial to the unit circle, and by the spacing of
	www.wavelet.org /phpBB2/viewtopic.php?t=3303 (132 words)

	Christophe Doche's home page (Site not responding. Last check: 2007-10-29)
		Furthermore if L is the length of the Rudin-Shapiro polynomial of order n, that is the sum of the absolute value of its coefficients i.e.
		A gp program to precisely compute the moments of the Rudin-Shapiro polynomial of order q even less than or equal to 32 and the moments of some generalized Rudin-Shapiro polynomials is available here [moments.gp.gz] as well as the tables of minimal polynomials for the recurrences and the first moments [data.gz] et [data32.gz]
		A table of redundant polynomials written in gp is also available.
	www.math.u-bordeaux.fr /~cdoche/indexeng.html (744 words)

	Home Page of Christophe Doche
		Zhang-Zagier heights of perturbed polynomials, proceedings of the XXIe Journees Arithmetiques.
		Moments of the Rudin-Shapiro Polynomials, with Laurent Habsieger, J. Fourier Analysis Appl.
		A gp program to precisely compute the moments of the Rudin-Shapiro polynomial of order q even less than or equal to 32 and the moments of some generalized Rudin-Shapiro polynomials is available here [moments.gp] as well as the tables of minimal polynomials for the recurrences and the first moments [data.gz] and [data32.gz]
	www.ics.mq.edu.au /~doche (256 words)

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