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...