A standard form for scattered linearized polynomials and properties of the related translation planes

Abstract

AbstractIn this paper, we present results concerning the stabilizer $$G_f$$ G f in $${{\,\mathrm{{GL}}\,}}(2,q^n)$$ GL ( 2 , q n ) of the subspace $$U_f=\{(x,f(x)):x\in \mathbb {F}_{q^n}\}$$ U f = { ( x , f ( x ) ) : x ∈ F q n } , f(x) a scattered linearized polynomial in $$\mathbb {F}_{q^n}[x]$$ F q n [ x ] . Each $$G_f$$ G f contains the $$q-1$$ q - 1 maps $$(x,y)\mapsto (ax,ay)$$ ( x , y ) ↦ ( a x , a y ) , $$a\in \mathbb {F}_{q}^*$$ a ∈ F q ∗ . By virtue of the results of Beard (Duke Math J, 39:313–321, 1972) and Willett (Duke Math J 40(3):701–704, 1973), the matrices in $$G_f$$ G f are simultaneously diagonalizable. This has several consequences: (i) the polynomials such that $$|G_f|>q-1$$ | G f | > q - 1 have a standard form of type $$\sum _{j=0}^{n/t-1}a_jx^{q^{s+jt}}$$ ∑ j = 0 n / t - 1 a j x q s + j t for some s and t such that $$(s,t)=1$$ ( s , t ) = 1 , $$t>1$$ t > 1 a divisor of n; (ii) this standard form is essentially unique; (iii) for $$n>2$$ n > 2 and $$q>3$$ q > 3 , the translation plane $$\mathcal {A}_f$$ A f associated with f(x) admits nontrivial affine homologies if and only if $$|G_f|>q-1$$ | G f | > q - 1 , and in that case those with axis through the origin form two groups of cardinality $$(q^t-1)/(q-1)$$ ( q t - 1 ) / ( q - 1 ) that exchange axes and coaxes; (iv) no plane of type $$\mathcal {A}_f$$ A f , f(x) a scattered polynomial not of pseudoregulus type, is a generalized André plane.