Square root Bound on the Least Power Non-residue using a Sylvester-Vandermonde Determinant - Forbes, Michael; Kayal, Neeraj; Mittal, Rajat; Saha, Chandan (2011)
We give a new elementary proof of the fact that the value of the least k ^th power nonresidue in an arithmetic progression [bn + c]n=0,1..., over a prime field Fp, is bounded by 7/ √ 5 · b · p p/k + 4b + c. Our proof is inspired by the so called Stepanov method, which involves bounding the size of the solution set of a system of equations by constructing a nonzero low degree auxiliary polynomial that vanishes with high multiplicity on the solution set. The proof uses basic algebra and number theory along with a determinant identity that generalizes both the Sylvester and the Vandermonde determinant.