Polynomial Pell’s equation

Webb, William A.; Yokota, Hisashi

doi:10.1090/s0002-9939-02-06934-4

Cited by 14 publications

(7 citation statements)

References 5 publications

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“…Equivalently, it is said that the triple (T n , (1 − x 2 ), U n ) is a solution to (polynomial) Pell's equation for every n ≥ 1. For more details on polynomial Pell's equation (originally Pell's equation is a topic in algebraic number theory), the interested reader is referred to [21,31]. Next, letting x → g(x) := (1 − x 2 ), and after normalization to pass to orthonormal polynomials, in summing up one obtains (5.10)…”

Section: 4mentioning

confidence: 99%

Positivity and Optimization: Beyond Polynomials

Lasserre

Putinar

2011

International Series in Operations Research &Amp; Management Science

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We briefly recall basics of the Moment-SOS hierarchy in polynomial optimization and the Christoffel-Darboux kernel (and the Christoffel function (CF)) in theory of approximation and orthogonal polynomials. We then (i) show a strong link between the CF and the SOS-based positive certificate at the core of the Moment-SOS hierarchy, and (ii) describe how the CDkernel provides a simple interpretation of the SOS-hierarchy of lower bounds as searching for some signed polynomial density (while the SOS-hierarchy of upper bounds is searching for a positive (SOS) density). This link between the CF and positive certificates, in turn allows us (i) to establish a disintegration property of the CF much like for measures, and (ii) for certain sets, to relate the CF of their equilibrium measure with a certificate of positivity on the set, for constant polynomials.

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Section: 4mentioning

confidence: 99%

Positivity and Optimization: Beyond Polynomials

Lasserre

Putinar

2011

International Series in Operations Research &Amp; Management Science

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“…Hence (37) is just (31) in Theorem 5.1 for the constant polynomial p(x) = 2n + 1 for all x on S = [−1, 1] = { x : g(x) ≥ 0}. So here the mysterious linear functional φ ∈ R[x] * 2n ∈ int(Q * n (g)) in Theorem 5.1 is the vector of moments µ = (µ j ) j≤2n up to degree 2n, of the Chebyshev measure µ = dx/π √ 1 − x 2 , which is the equilibrium measure of S.…”

Section: 1mentioning

confidence: 99%

The Christoffel function: Applications, connections and extensions

Lasserre

2024

NACO

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We provide an introduction to the Christoffel function (CF), a well-known (and old) tool from the theory of approximation and orthogonal polynomials. We then describe how it provides a simple and easy-to-use tool to address some problems in data analysis and mining, and in approximation and interpolation of discontinuous functions. Finally we also reveal some surprising links of the CF with seemingly different disciplines, including polynomial optimization, positivity certificates, and equilibrium measures of compact sets.

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“…Then, using the method outlined in Section 1, one can prove easily the following analogue of Theorem 2 in [2] for the sign changes of λ function at rational points f (r), r ∈ Q, namely: either λ(f (r)) is constant for all rational numbers r greater than the largest real root of g(x) − x or it changes sign infinitely many often. The question of finding all solutions of the composition equation in integer polynomials f (x), g(x), and h(x) is closely related to the solution of the polynomial Pell equations in Z[x], see [9], [10], [14]. This does not seem to be easy.…”

Section: Rational and Integer Examplesmentioning

confidence: 99%

On the equation $f(g(x)) =f(x)h^m(x)$ for composite polynomials

Ganguli¹,

Jankauskas²

2012

Preprint

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In this paper we solve the equation f (g(x)) = f (x)h m (x) where f (x), g(x) and h(x) are unknown polynomials with coefficients in an arbitrary field K, f (x) is nonconstant and separable, deg g 2, the polynomial g(x) has non-zero derivative g ′ (x) = 0 in K[x] and the integer m 2 is not divisible by the characteristic of the field K. We prove that this equation has no solutions if deg f 3. If deg f = 2, we prove that m = 2 and give all solutions explicitly in terms of Chebyshev polynomials. The diophantine applications for such polynomials f (x), g(x), h(x) with coefficients in Q or Z are considered in the context of the conjecture of Cassaign et. al on the values of Louiville's λ function at points f (r), r ∈ Q.

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Polynomial Pell’s equation

Cited by 14 publications

References 5 publications

Positivity and Optimization: Beyond Polynomials

Positivity and Optimization: Beyond Polynomials

The Christoffel function: Applications, connections and extensions

On the equation $f(g(x)) =f(x)h^m(x)$ for composite polynomials

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