Real Analysis Exchange

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

Krzysztof Chris Ciesielski

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

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

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