More translation planes and semifields from Dembowski-Ostrom polynomials (Q1951217)

Let \(F=\mathrm{GF}(q^n)\), where \(q\) is an odd prime power and \(n\) is odd, and let \(\zeta \in \mathrm{GF}(q)\) be a non-square. In a recent paper, P. Müller and the author studied Dembowsky-Ostrom polynomials of the form \(P(X)=L(X)X\), with \(L(X)=\sum_{i=0}^{n-1} a_iX^{q^i}\), such that \((1)\) \(x \mapsto L(x)\) is bijective, \((2)\) \(|P(F^*)|=\frac{q^n-1}{2}\), and \((3)\) \(P(F^*)\cap \zeta P(F^*) =\emptyset\). They also investigated their relation with translation planes. In this article, the author constructs new translation planes of order \(q^n\) using pairs of Dembowsky-Ostrom polynomials satisfying \((1)\), \((2)\), and \((3)\). These translation planes admit cyclic groups of order \(q^n-1\) having orbits of length \(1,1,\frac{q^n-1}{2},\frac{q^n-1}{2}\) on the line at infinity. Also, using the same polynomials, semifields of order \(q^{2n}\) are constructed and their nuclei are computed.