{"id":354569,"date":"2017-01-18T09:55:11","date_gmt":"2017-01-18T17:55:11","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=354569"},"modified":"2018-10-16T19:57:32","modified_gmt":"2018-10-17T02:57:32","slug":"square-root-bound-least-power-non-residue-using-sylvester-vandermonde-determinant","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/square-root-bound-least-power-non-residue-using-sylvester-vandermonde-determinant\/","title":{"rendered":"Square root Bound on the Least Power Non-residue using a Sylvester-Vandermonde Determinant"},"content":{"rendered":"

We give a new elementary proof of the fact that the value of the least k ^th<\/em> power nonresidue in an arithmetic progression {bn + c}n=0,1<\/em>…, over a prime field Fp, is bounded by 7\/ \u221a 5 \u00b7 b \u00b7 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.<\/p>\n","protected":false},"excerpt":{"rendered":"

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\/ \u221a 5 \u00b7 b \u00b7 p p\/k + 4b + c. Our proof is inspired by the so called Stepanov 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