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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.

##### MSC:
 11T06 Polynomials over finite fields