On the cardinality of value set of polynomials with coefficients in a finite field (Q2366793)

Let \(F_ q\) denote the finite field of order \(q\), a prime power, and \(f(x)\) a polynomial of degree \(d\) over \(F_ q\). Let \(V_ f=\{f(x)\): \(x\in F_ q\}\) denote the value set of \(F(x)\) and \(| V_ f|\) its cardinality. Various lower bounds on \(V_ f\) have been determined. It is shown in this paper that if \((d,q)=1\), \(d^ 4<q\) and the multiplicative order of \(q\) modulo \(p_ i^{a_ i}\) is \(p_ i^{a_ i}-p_ i^{a_ i-1}\) for all prime powers \(p_ i^{a_ i}\| d\), then \[ | V_ f| \geq {q \over {1+\sum_{D| d} \varphi(D)/\text{lcm}(\varphi(p_ 1^{b_ 1}),\cdots,\varphi(p_ r^{b_ r}))}} \] where \(D=p_ 1^{b_ 1} \cdots p_ r^{b_ r}\) and \(\varphi(D)\) denotes the Euler phi function.