On proportion polynomials (Q1281738)

The notion of proportion functions was introduced by this reviewer [Problem P272, Aequat. Math 29, p. 100 (1985)] motivated by some architectural problems. They are mappings \( f\) from \(\mathbb{R}^{++} \times \mathbb{R}^{++}\) into \([1, \infty) \) such that \( f (x,y) = f({{y^2}\over x},y) \), \( f(x,y) = f (y,x)\) and \( f(x,x) = 1 \). Characterizations have been found by W. Benz and Z. Moszner. The author of this paper presents interesting results showing all proportion polynomials of maximum degree three and proportion polynomials of class \( C^1 \) of maximum degree five. Moreover a class of weak proportion polynomials is completely characterized. Arguments are clear and inspiring.