Rank-Two Frobenius-Linearized Normal Forms and Orthoderivative Dual Coordinates in Quadratic APN Maps

• URL: http://arxiv.org/abs/2608.11939v1
• Date: Wed, 12 Aug 2026 11:27:30 GMT
• Title: Rank-Two Frobenius-Linearized Normal Forms and Orthoderivative Dual Coordinates in Quadratic APN Maps
• Abstract summary: We classify binary-linear two-term Frobenius-linearized operators $L(Y)=AY+BY$ on $K3$.<n>For pure $$-quadratic almost perfect nonlinear maps, this identifies the orthoderivative by $_F(X)T F(X)=1$.<n>The triprojective construction of Gologlusch and the cubic norm-twist construction of Li, Zhou, Li, and Qu provide two realizations.
• Score: 1.5250946159271779
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We classify binary-linear two-term Frobenius-linearized operators $L(Y)=AY^σ+BY$ on $K^3$, where $K$ is a finite extension of $\mathbb{F}_2$ and $σ$ is a fixed nontrivial Frobenius automorphism of $K$ with fixed field $\mathbb{F}_2$. Under a coefficient-rank and binary-kernel condition, if $A$ and $B$ both have $K$-rank two and $L$ has a one-dimensional kernel over $\mathbb{F}_2$, then invertible $K$-linear input and output changes reduce $L$, for this fixed $σ$, to the canonical model $(α,β,γ)\mapsto(α^σ+α,β^σ,γ)$. The proof constructs the coordinate frames from the two coefficient-kernel directions and the binary kernel. In these coordinates, the first dual output row is exactly the unique nonzero trace-adjoint normal, with an exact $K$-valued normalization. For pure $σ$-quadratic almost perfect nonlinear maps, this identifies the orthoderivative by $π_F(X)^T F(X)=1$; in odd extension degree it also yields permutation behavior and a bijection from the projective plane to its dual. The triprojective construction of Gologlu and Kolsch and the cubic norm-twist construction of Li, Zhou, Li, and Qu provide two realizations arising from different algebraic constructions. The triprojective case further admits a determinant factorization and a complete dual frame, whereas the norm-twist realization shows that the pure-map consequences do not follow from the operator theorem alone. A natural Gold representation has coefficient-rank pair $(3,3)$, delimiting the rank-two subclass. The normal form also supplies exact extension-field labels for known component-radical and Walsh-support relations.

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