# 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 $$\deg f. If $$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?