A divided-difference characterization of polynomials over finite fields of characteristic two (Q2123711)

Suppose that \(\mathbb{F}\) is a field and \(f, h:\mathbb{F}\to \mathbb{F}\) satisfy \( f [x_1, \dots , x_ n] = h (x_ 1 + \dots + x_ n)\), whenever \(x_1, \dots, x_n\) are distinct elements of \(\mathbb{F}\) and \(f [x_1, \dots, x_ n]\) denotes the divided difference of \(f\) at the distinct points \(x_1, \dots, x_n\). \textit{R. O. Davies} and \textit{G. Rousseau} [Aequationes Math. 55, No. 1--2, 73--78 (1998; Zbl 0892.39020)] proved that for an arbitrary field \(\mathbb{F}\) not of characteristic \(2\) and arbitrary \(n\geq 2\), \(f\) is equal to a polynomial of degree at most \(n\) over \(\mathbb{F}\). In this paper, the authors prove that if \(\mathbb{F}\) is a finite field of characteristic \(2\), and \(n\geq 3\), then \(f\) is equal to a polynomial of degree at most \(n\) over \(\mathbb{F}\).