Displaying similar documents to “On the integer solutions of exponential equations in function fields”

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...

On coefficient valuations of Eisenstein polynomials

Matthias Künzer, Eduard Wirsing (2005)

Journal de Théorie des Nombres de Bordeaux

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Let p ≥ 3 be a prime, let n > m ≥ 1 . Let π n be the norm of ζ p n - 1 under C p - 1 , so that ℤ ( p ) [ π n ] | ℤ ( p ) is a purely ramified extension of discrete valuation rings of degree p n - 1 . The minimal polynomial of π n over ℚ ( π m ) is an Eisenstein polynomial; we give lower bounds for its coefficient valuations at π m . The function field analogue, as introduced by Carlitz and Hayes, is studied as well.

Comments on the height reducing property

Shigeki Akiyama, Toufik Zaimi (2013)

Open Mathematics

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A complex number α is said to satisfy the height reducing property if there is a finite subset, say F, of the ring ℤ of the rational integers such that ℤ[α] = F[α]. This property has been considered by several authors, especially in contexts related to self affine tilings and expansions of real numbers in non-integer bases. We prove that a number satisfying the height reducing property, is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one,...