Quadratic forms of codimension 2 over finite fields containing \(\mathbb F_4\) and Artin-Schreier type curves (Q765811)

Let \(\mathbb{F}_{q^k}/\mathbb{F}_q\) be an extension of finite fields of characteristic two. \textit{R. W. Fitzgerald} [Finite Fields Appl. 11, No. 2, 165--181 (2005; Zbl 1140.11331)] proved that every quadratic form \(Q: K\to F\) can be expressed, in an essentially unique way, as \(Q(x)=\text{Tr}_{K/F}(xR(x))\), where \(R(x)=\sum_{i=0}^{\lfloor k/2\rfloor} \varepsilon_i x^{q^i}\) is a linearized polynomial with coefficients in \(\mathbb{F}_{q^k}\). Also, the linear polynomials \(R\) with coefficients \(0\) or \(1\) where the form has a codimension 2 radical were characterized. In another paper, Fitzgerald extended this characterization to arbitrary linear polynomials \(R\) but assuming that the ground field was \(\mathbb{F}_q=\mathbb{F}_2\) [Finite Fields Appl. 13, No. 4, 778--792 (2007; Zbl 1137.11026)]. In the paper under review, the authors characterize the linear polynomials leading to quadratic forms with a codimension \(2\) radical, assuming that \(q\) is a square and \(\varepsilon_0,\varepsilon_1,\varepsilon_2\) belong to \(\mathbb{F}_4\). In all these cases, they determine the invariant \(\Lambda(Q)\in\left\{0,1,-1\right\}\) such that the number of zeros of \(Q\) is \(q^{k-1}+\Lambda(Q)(q-1)q^{k-2}\). This allows to control the number of rational points of the Artin-Schreier curve \(y^q+y=xR(x)\), leading to new results in the classification of maximal and minimal curves over \(\mathbb{F}_{q^k}\).