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

Charles N. Delzell (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

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 natural numbers;...

Extensions of the Bloch–Pólya theorem on the number of real zeros of polynomials

Tamás Erdélyi (2008)

Journal de Théorie des Nombres de Bordeaux

We prove that there are absolute constants c 1 > 0 and c 2 > 0 such that for every { a 0 , a 1 , ... , a n } ⊂ [ 1 , M ] , 1 ≤ M ≤ exp ( c 1 n 1 / 4 ) , there are b 0 , b 1 , ... , b n ∈ { - 1 , 0 , 1 } such that P ( z ) = ∑ j = 0 n b j a j z j has at least c 2 n 1 / 4 distinct sign changes in ( 0 , 1 ) . This improves and extends earlier results of Bloch and Pólya.

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