<abstract xmlns="http://www.w3.org/1999/xhtml">
<p>In this paper we consider a generalized polynomial <italic>f</italic>: ℝ → ℝ of degree two that satisfies the additional equation <italic>f</italic>(<italic>x</italic>)<italic>f</italic>(<italic>y</italic>) = 0 for the pairs (<italic>x, y</italic>) ∈ <italic>D</italic>, where <italic>D</italic> ⊆ ℝ<sup>2</sup> is given by some algebraic condition. In the particular cases when there exists a positive rational <italic>m </italic>fulfilling
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2023-0017_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>D</mi><mo>=</mo><mrow><mo>{</mo> <mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>∈</mo><msup><mrow><mi>ℝ</mi></mrow><mn>2</mn></msup><mo>|</mo><msup><mrow><mi>x</mi></mrow><mn>2</mn></msup><mo>-</mo><mi>m</mi><msup><mrow><mi>y</mi></mrow><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow> <mo>}</mo></mrow><mo>,</mo></mrow></math>
<tex-math>D = \left\{ {\left( {x,y} \right) \in \mathbb{R}{^2}|{x^2} - m{y^2} = 1} \right\},</tex-math>
</alternatives>
</disp-formula>
we prove that <italic>f</italic>(<italic>x</italic>) = 0 for all <italic>x</italic> ∈ ℝ.</p>
</abstract>

In this paper we consider a generalized polynomial f: ℝ → ℝ of degree two that satisfies the additional equation f(x)f(y) = 0 for the pairs (x, y) ∈ D, where D ⊆ ℝ2 is given by some algebraic condition. In the particular cases when there exists a positive rational m fulfilling



D={ (x,y)∈ℝ2|x2-my2=1 },
D = \left\{ {\left( {x,y} \right) \in \mathbb{R}{^2}|{x^2} - m{y^2} = 1} \right\},


we prove that f(x) = 0 for all x ∈ ℝ.

An Alternative Equation for Generalized Polynomials of Degree Two

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"An Alternative Equation for Generalized Polynomials of Degree Two"}

In this paper we consider a generalized polynomial f: ℝ → ℝ of degree two that satisfies the additional equation f(x)f(y) = 0 for the pairs (x, y) ∈ D, where D...

An Alternative Equation for Generalized Polynomials of Degree Two

Published Online: Oct 26, 2023

Page range: -

Received: Apr 25, 2023

Accepted: Oct 03, 2023

DOI: https://doi.org/10.2478/amsil-2023-0017

Keywords
additive functions, quadratic functions, generalized polynomials, alternative equation

© 2023 Zoltán Boros, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

An Alternative Equation for Generalized Polynomials of Degree Two

Published Online: Oct 26, 2023

Page range: -

Received: Apr 25, 2023

Accepted: Oct 03, 2023

DOI: https://doi.org/10.2478/amsil-2023-0017

Keywordsadditive functions, quadratic functions, generalized polynomials, alternative equation

© 2023 Zoltán Boros, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
additive functions, quadratic functions, generalized polynomials, alternative equation