Planarity of products of two linearized polynomials. (English) Zbl 1271.11112

Summary: Let \(L_1(x)\) and \(L_2(x)\) be linearized polynomials over \(\mathbb F_{q^n}\). We give conditions when the product \(L_1(x)\cdot L_2(x)\) defines a planar mapping on \(\mathbb F_{q^n}\). For a polynomial \(L\) over \(\mathbb F_{q^n}\), let \(M(L)=\{\alpha \in \mathbb F_{q^n}: L(x)+\alpha \cdot x\}\) is bijective on \(\mathbb F_{q^n}\). We show that the planarity of the product \(L_1(x)\cdot L_2(x)\) is linked with the set \(M(L)\) of a suitable linearized polynomial \(L\). We use this relation to describe families of such planar mappings as well as to obtain nonexistence results.


11T06 Polynomials over finite fields
12E10 Special polynomials in general fields
Full Text: DOI


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