{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article20515","name":"On the Degree of Polynomials Computing Square Roots Mod p","abstract":"For an odd prime p, we say f(X) \u2208 F_p[X] computes square roots in F_p if, for all nonzero perfect squares a \u2208 F_p, we have f(a)\u00b2 = a.\r\nWhen p \u2261 3 mod 4, it is well known that f(X) = X^{(p+1)\/4} computes square roots. This degree is surprisingly low (and in fact lowest possible), since we have specified (p-1)\/2 evaluations (up to sign) of the polynomial f(X). On the other hand, for p \u2261 1 mod 4 there was previously no nontrivial bound known on the lowest degree of a polynomial computing square roots in F_p.\r\nWe show that for all p \u2261 1 mod 4, the degree of a polynomial computing square roots has degree at least p\/3. Our main new ingredient is a general lemma which may be of independent interest: powers of a low degree polynomial cannot have too many consecutive zero coefficients. The proof method also yields a robust version: any polynomial that computes square roots for 99% of the squares also has degree almost p\/3. \r\nIn the other direction, Agou, Deligl\u00e9se, and Nicolas [Agou et al., 2003] showed that for infinitely many p \u2261 1 mod 4, the degree of a polynomial computing square roots can be as small as 3p\/8.","keywords":["Algebraic Computation","Polynomials","Computing Square roots","Reed-Solomon Codes"],"author":[{"@type":"Person","name":"Kedlaya, Kiran S.","givenName":"Kiran S.","familyName":"Kedlaya","email":"mailto:kedlaya@ucsd.edu","sameAs":"https:\/\/orcid.org\/0000-0001-8700-8758","affiliation":"Department of Mathematics, University of California San Diego, La Jolla, CA, USA","funding":"Research supported by NSF grant DMS-2053473, the UC San Diego Warschawski Professorship, the Simons Fellows in Mathematics program of the Simons Foundation (2023-24 academic year)."},{"@type":"Person","name":"Kopparty, Swastik","givenName":"Swastik","familyName":"Kopparty","email":"mailto:swastik.kopparty@utoronto.ca","sameAs":"https:\/\/orcid.org\/0000-0003-2704-8808","affiliation":"Department of Mathematics and Department of Computer Science, University of Toronto, Canada","funding":"Research supported by an NSERC Discovery Grant."}],"position":25,"pageStart":"25:1","pageEnd":"25:14","dateCreated":"2024-07-15","datePublished":"2024-07-15","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kedlaya, Kiran S.","givenName":"Kiran S.","familyName":"Kedlaya","email":"mailto:kedlaya@ucsd.edu","sameAs":"https:\/\/orcid.org\/0000-0001-8700-8758","affiliation":"Department of Mathematics, University of California San Diego, La Jolla, CA, USA","funding":"Research supported by NSF grant DMS-2053473, the UC San Diego Warschawski Professorship, the Simons Fellows in Mathematics program of the Simons Foundation (2023-24 academic year)."},{"@type":"Person","name":"Kopparty, Swastik","givenName":"Swastik","familyName":"Kopparty","email":"mailto:swastik.kopparty@utoronto.ca","sameAs":"https:\/\/orcid.org\/0000-0003-2704-8808","affiliation":"Department of Mathematics and Department of Computer Science, University of Toronto, Canada","funding":"Research supported by an NSERC Discovery Grant."}],"copyrightYear":"2024","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.CCC.2024.25","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":"https:\/\/arxiv.org\/abs\/1101.4605","isPartOf":{"@type":"PublicationVolume","@id":"#volume20488","volumeNumber":300,"name":"39th Computational Complexity Conference (CCC 2024)","dateCreated":"2024-07-15","datePublished":"2024-07-15","editor":{"@type":"Person","name":"Santhanam, Rahul","givenName":"Rahul","familyName":"Santhanam","email":"mailto:rahul.santhanam@cs.ox.ac.uk","sameAs":"https:\/\/orcid.org\/0000-0002-8716-6091","affiliation":"University of Oxford, UK"},"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article20515","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume20488"}}}