An alternative equation for generalized monomials

In this paper we consider a generalized monomial or polynomial f:R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f : \mathbb {R}\rightarrow \mathbb {R}$$\end{document} that satisfies the additional equation f(x)f(y)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f(x) f(y) = 0 $$\end{document} for the pairs (x,y)∈D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ (x,y) \in D \,$$\end{document}, where D⊆R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ D \subseteq {\mathbb {R}}^{2} $$\end{document} is given by some algebraic condition. In the particular cases when f is a generalized polynomial and there exist non-constant regular polynomials p and q that fulfill D={(p(t),q(t))|t∈R}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D = \{\, (p(t),q(t)) \,\vert \, t \in \mathbb {R}\,\} \end{aligned}$$\end{document}or f is a generalized monomial and there exists a positive rational m fulfilling D={(x,y)∈R2|x2-my2=1},\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D = \{\, (x,y) \in {\mathbb {R}}^{2} \,\vert \, x^2 - m y^2 = 1 \,\}, \end{aligned}$$\end{document}we prove that f(x)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f(x) = 0 $$\end{document} for all x∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ x \in \mathbb {R}\,$$\end{document}.


Introduction
Let R , Q , and N denote the set of all real numbers, rationals, and positive integers, respectively. We call a function f : R → R additive if f (x + y) = f (x) + f (y) holds for all x, y ∈ R . The function f is called Q-homogeneous if the equation f (qx) = qf (x) is fulfilled by every q ∈ Q and x ∈ R . As it is also well-known [6, Theorem 5.2.1], if f : R → R is additive, then f is Q-homogeneous as well. We define the following sets: Moreover, if p and q are regular, non-constant, real polynomials, while m is a positive real number, we shall also consider the sets Kominek, L. Reich and J. Schwaiger [5] investigated additive real functions that satisfy the additional equation for every (x, y) ∈ D, considering various subsets D of R 2 . In several cases, involving D = R p,q and D = S 2 , they obtained f (x) = 0 for every x ∈ R. Their result for D = S 2 was extended by Z. Boros and W. Fechner [1] to the situation when f is a generalized polynomial. On the other hand, P. Kutas [7] has recently established the existence of a non-zero additive function f : R → R fulfilling (1) for all (x, y) ∈ S 0 . The case of bounded f (x)f (y) on S 2 was investigated by these authors [2]. The purpose of the present paper is to involve the case D = S 1,m into this research for every positive rational m. We note that, in some sense, S 1,1 = S 1 is on a half way from S 0 to S 2 , as it is geometrically analogous to S 0 and algebraically analogous to S 2 . Moreover, motivated by [1, Theorem 1], we wish to extend the investigation of the cases D = R p,q and D = S 1,m for a generalized polynomial or monomial f , respectively.

Concepts and lemmas
Let n ∈ N . A function F : R n → R is called n-additive if F is additive in each of its variables. Clearly, an n-additive function is also Q-homogeneous in each variable.
Given a function F : R n → R, by the diagonalization (or trace) of F we understand the function f : R → R arising from F by putting all the variables (from R) equal: is also clear that the set of all monomials of degree n is a real linear space with respect to the pointwise operations for any non-negative integer n.
If f is a finite sum of generalized monomials, then f is called a generalized polynomial.
For more information concerning these notions the reader is referred to the monograph by M. Kuczma [6,Chapter 15.9] as well as to the short introduction in [1].
In order to make use of the already mentioned Q-homogeneity property of n-additive functions, in our arguments we shall repeatedly apply the following observation. If a regular real polynomial P (s) equals zero for infinitely many distinct values of the variable s, then it is identically zero, i.e., the coefficient of s k equals zero for every non-negative integer k (up to the degree of P ). Clearly, it follows from the fact that for a not identically zero polynomial P the equation P (s) = 0 is satisfied only by a finite number of distinct values of s. This idea is explicitly stated in [3, Lemma 1]. The application of this idea in the theory of functional equations goes back to the paper by Nishiyama and Horinouchi [8].
We shall also need to verify the following statements.
Proof. Due to our assumptions, there exist a positive integer n and k-additive for every . . , n). According to the hypothesis, I ⊆ R is an interval with positive length. From the density of Q in R we can see that for any real number x = 0 there exist infinitely many r ∈ Q such that rx ∈ I and thus We have just obtained a polynomial of degree (at most) n with infinitely many rational zeroes. This implies that the polynomial is identically zero, . Therefore, f vanishes on R . Proof. Let j be a positive integer such that f is a monomial of degree j, i.e., f is the diagonalization of the j-additive mapping A : R j → R . Moreover, let n be a non-negative integer and a k ∈ R (k = 0, 1, . . . .n) such that For any fixed non-negative integers k l ∈ { 0, , 1, , . . . , n } (l = 1, 2, . . . , j), let s = j l=1 k l and where any empty product equals 1 (i.e., for k l = 0 we have only a k l in the l-th entry of A). Due to the distributivity of multiplication of real numbers and the j-additivity of A , G is s-additive and Thus f • p is a finite sum of generalized monomials, hence it is a generalized polynomial.

Main results
Now we can establish our main theorems. The first one involves two nonconstant regular polynomials with possibly different degrees.
for every x ∈ R , then f (x) = 0 identically. as well. Therefore the functional equation (3) implies that either f (p(x)) = 0 identically or f (q(x)) = 0 identically. Due to our assumptions the ranges p(R) and q(R) are unbounded intervals, hence f vanishes on an unbounded interval. Applying Lemma 2.1 we obtain that f (x) = 0 for all x ∈ R .
Our second theorem involves particular hyperbolas. The major tool in our arguments is obtained by a family of linear transformations that leave such a hyperbola invariant. Proof. It is obtained by a straightforward calculation.  Proof. Given a generalized monomial f , we can associate a positive integer k and a k-additive and symmetric functional A : R k → R with f in such a way that for all x ∈ R . Now, let x ∈ R such that x ≥ 1 . Then there exists 0 ≤ y ∈ R such that x 2 − my 2 = 1 . If α, β are rational numbers such that α 2 − mβ 2 = 1 , Lemma 3.2 and our assumptions on f imply Next, let us denote Due to Eq. (5), for every pair of rationals (α, β) fulfilling α 2 − mβ 2 = 1 , at least one of the foregoing expressions is equal to zero.
What is more, we can find infinitely many distinct pairs (α j , β j ) such that α 2 j − mβ 2 j = 1 and both α j and β j are rationals, so let us take for j ∈ N such that mj 2 = 1 .
Multiplying both equations by (mj 2 − 1) k and introducing the functions we have P (j) = 0 orP (j) = 0 for each integer j ∈ N m . Hence either P orP has infinitely many zeros. On the other hand, both P andP are polynomials of degree not greater than 2k . Therefore, one of them has to be identically equal to 0 . So either i.e., f (x) = 0 (here i denotes the imaginary unit as polynomials with real coefficients can be considered as polynomials over the complex number field as well).
We have thus proved f (x) = 0 for every real number x ≥ 1 . Hence, applying Lemma 2.1, we obtain that f (x) = 0 for all x ∈ R . Proof. Let u and w be real numbers fulfilling the condition u 2 − a 2 b 2 w 2 = 1 . Moreover, let g(t) = f (at) for all t ∈ R . Clearly, then g is a generalized monomial as well. For x = au and y = aw we have Therefore g satisfies the assumptions in Theorem 3.4 with m = a 2 b 2 , hence g is identically equal to zero, which yields f (x) = g(x/a) = 0 for every x ∈ R as well.
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