Real Analysis Exchange

Minimal Degrees of Genocchi-Peano Functions: Calculus Motivated Number Theoretical Estimates

Krzysztof Chris Ciesielski

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

A rational function of the form \(\frac{x_1^{\alpha_1} x_2^{\alpha_2} \cdots x_n^{\alpha_n}}{x_1^{\beta_1} + x_2^{\beta_2}+\cdots + x_n^{\beta_n}}\) is a Genocchi-Peano example, GPE, provided it is discontinuous, but its restriction to any hyperplane is continuous. We show that the minimal degree \(D(n)\) of a GPE of \(n\)-variables equals \(2\left\lfloor \frac{e^2}{e^2-1} n \right\rfloor+2i\) for some \(i\in\{0,1,2\}\). We also investigate the minimal degree \(D_b(n)\) of a bounded GPE of \(n\)-variables and note that \(D(n)\leq D_b(n)\leq n(n+1)\). Finding better bounds for the numbers \(D_b(n)\) remains an open problem.

Article information

Source
Real Anal. Exchange, Volume 43, Number 2 (2018), 281-292.

Dates
First available in Project Euclid: 27 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.rae/1530064961

Digital Object Identifier
doi:10.14321/realanalexch.43.2.0281

Mathematical Reviews number (MathSciNet)
MR3942578

Zentralblatt MATH identifier
06924889

Subjects
Primary: 11A25: Arithmetic functions; related numbers; inversion formulas
Secondary: 26B05: Continuity and differentiation questions

Keywords
separate continuity hyperplane continuity smallest degree Genocchi-Peano examples

Citation

Ciesielski, Krzysztof Chris. Minimal Degrees of Genocchi-Peano Functions: Calculus Motivated Number Theoretical Estimates. Real Anal. Exchange 43 (2018), no. 2, 281--292. doi:10.14321/realanalexch.43.2.0281. https://projecteuclid.org/euclid.rae/1530064961


Export citation