Linearized polynomials and permutation polynomials of finite fields (Q1803912)

Let \(F_ q\) denote the finite field of order \(q=p^ m\) with \(m\geq 1\) and \(p\) is prime. A polynomial \(f\in F_ q[x]\) is called a permutation polynomial (PP) of \(F_ q\) if the mapping induced on \(F_ q\) by \(f\) is a bijection. Among other results, the authors prove that if \(f\in F_ q[x]\) with \(\deg(f)<q\) is such that \(f(x)+cx\) is a PP for at least \(\lfloor q/2\rfloor\) values of \(c\in F_ q\), then \(f(x)+cx\) is a PP for at least \(q-(q-1)/(p-1)\) values of \(c\in F_ q\) and moreover, \(f(x)=ax+g(x^ p)\) for some \(a\in F_ q\) and \(g\in F_ q[x]\). The authors also conjecture that if \(f\in F_ q[x]\) is such that \(f(x)+cx\) is a PP for at least \(\lfloor q/2\rfloor\) values of \(c\in F_ q\), then \(f(x)-f(0)\) is a linearized \(p\)-polynomial over \(F_ q\), i.e. each of the terms of \(f\) has degree equal to a power of \(p\). The above result verifies the conjecture when \(q=p\) and the authors prove using the theory of Gauss and Jacobi sums over finite fields that the conjecture is also true whenever \(f(x)=x^ e\) is a monomial. Such results concerning polynomials \(f\) with the property that \(f(x)+cx\) is a PP for many values of \(c\in F_ q\) have applications in combinatorics, in particular in the study of sets of mutually orthogonal latin squares.