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The factorable core of polynomials over finite fields. (English) Zbl 0728.11065
Let \(f(x)\) denote a polynomial over a finite field \(\mathbb F_q\) and define \(\phi_ f(x,y)=f(y)-f(x)\). The polynomial \(\phi_f\) has frequently been used to study questions concerning the value set of \(f\). The author proves the existence of a polynomial \(F\) over \(\mathbb F_q\) of the form \(F=L^r\), where \(L\) is an affine linearized polynomial over \(\mathbb F_q\), such that \(f=g(F)\) for some polynomial \(g\), and the part of \(\phi_f\) that splits completely into linear factors over the algebraic closure of \(\mathbb F_q\) is exactly \(\phi_F\). Following some early terminology of L. Carlitz [Duke Math. J. 2, 660–670 (1936; Zbl 0015.29301)] and A. F. Long [Duke Math. J. 34, 281–291 (1967; Zbl 0154.03604)] \(F\) is called “factorable” and the “factorable core” of \(f\). This result sheds light on an aspect of work of D. R. Hayes [Duke Math. J. 34, 293–305 (1967; Zbl 0163.05202)] and Daqing Wan [J. Aust. Math. Soc., Ser. A 43, 375–384 (1987; Zbl 0635.12011)] on the existence of permutation polynomials of even degree over \(\mathbb F_q\). Some related results on value sets, including the exhibition of a certain class of permutation polynomials, are also briefly discussed.

11T06 Polynomials over finite fields