A special case of the polynomial Bézout’s identity: bounding the co-factors

Let $ F$ be a field of prime order $ p$ . Suppose that $ f\in F[x]$ is a non-zero polynomial of degree $ \deg f<p$ . If $ f$ does not have multiple roots, then there exist polynomials $ P,Q\in F[x]$ such that $ $ Pf+Qf’=1.$ $ What is the smallest possible degree of $ P$ in this representation, and how it depends on $ f$ ?

Denoting this smallest possible degree by $ \nu(f)$ , some basic observations are:

  • $ \nu(cf)=\nu(f)$ for any $ c\in F^\times$ ;
  • if $ g(x)=f(cx+b)$ , then $ \nu(g)=\nu(f)$ for any $ b\in F$ and $ c\in F^\times$ ;
  • $ \nu(f)=0$ if and only if $ f(x)=a(x-b)^d+c$ with $ a,b\in F$ and $ c\in F^\times$ .

Is it possible to classify those polynomials $ f$ with $ \nu(f)<10$ ? With $ \nu(f)<\deg f$ ? Does $ \nu$ have any special properties allowing one to estimate or easily compute it?