Displaying similar documents to “A polynomial of degree four not satisfying Rolle’s Theorem in the unit ball of l 2 ”

On generalized “ham sandwich” theorems

Marek Golasiński (2006)

Archivum Mathematicum

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In this short note we utilize the Borsuk-Ulam Anitpodal Theorem to present a simple proof of the following generalization of the “Ham Sandwich Theorem”: Let A 1 , ... , A m ⊆ ℝ n be subsets with finite Lebesgue measure. Then, for any sequence f 0 , ... , f m of ℝ -linearly independent polynomials in the polynomial ring ℝ [ X 1 , ... , X n ] there are real numbers λ 0 , ... , λ m , not all zero, such that the real affine variety { x ∈ ℝ n ; λ 0 f 0 ( x ) + ⋯ + λ m f m ( x ) = 0 } simultaneously bisects each of subsets A k , k = 1 , ... , m . Then some its applications are studied.

Vieta’s Formula about the Sum of Roots of Polynomials

Artur Korniłowicz, Karol Pąk (2017)

Formalized Mathematics

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In the article we formalized in the Mizar system [2] the Vieta formula about the sum of roots of a polynomial anxn + an−1xn−1 + ··· + a1x + a0 defined over an algebraically closed field. The formula says that [...] x1+x2+⋯+xn−1+xn=−an−1an x 1 + x 2 + ⋯ + x n - 1 + x n = - a n - 1 a n , where x1, x2,…, xn are (not necessarily distinct) roots of the polynomial [12]. In the article the sum is denoted by SumRoots.

Extension of the Two-Variable Pierce-Birkhoff conjecture to generalized polynomials

Charles N. Delzell (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

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Let h : ℝ n → ℝ be a continuous, piecewise-polynomial function. The Pierce-Birkhoff conjecture (1956) is that any such h is representable in the form sup i inf j f i j , for some finite collection of polynomials f i j ∈ ℝ [ x 1 , ... , x n ] . (A simple example is h ( x 1 ) = | x 1 | = sup { x 1 , - x 1 } .) In 1984, L. Mahé and, independently, G. Efroymson, proved this for n ≤ 2 ; it remains open for n ≥ 3 . In this paper we prove an analogous result for “generalized polynomials” (also known as signomials), i.e., where the exponents are allowed to be arbitrary real numbers, and not just...

An Inequality for Trigonometric Polynomials

N. K. Govil, Mohammed A. Qazi, Qazi I. Rahman (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

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The main result says in particular that if t ( ζ ) : = ∑ ν = - n ⁿ c ν e i ν ζ is a trigonometric polynomial of degree n having all its zeros in the open upper half-plane such that |t(ξ)| ≥ μ on the real axis and cₙ ≠ 0, then |t’(ξ)| ≥ μn for all real ξ.

Restricted partitions.

Jakimczuk, Rafael (2004)

International Journal of Mathematics and Mathematical Sciences

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