The sum of square roots problem over integers is the task of deciding the sign of a non-zero sum, S =Pn i=1 \u03b4i \u00b7\u221aai, where \u03b4i \u2208 {+1,\u22121} and ai\u2019s are positive integers that are upperbounded by N (say). A fundamental open question in numerical analysis and computational geometry is whether |S|\u22651\/2(n\u00b7log N)O(1) when S 6= 0. We study a formulation of this problem over polynomials: Given an expression S =Pn i=1 ci \u00b7pfi(x), where ci\u2019s belong to a \ufb01eld of characteristic 0 and fi\u2019s are univariate polynomials with degree bounded by d and fi(0)6= 0 forall i, is it true that the minimum exponent of x which has a nonzero coe\ufb03cient in the power series S is upper bounded by (n\u00b7d)O(1), unless S = 0? We answer this question a\ufb03rmatively. Further, we show that this result over polynomials can be used to settle (positively) the sum of square roots problem for a special class of integers: Suppose each integer ai is of the form, ai = Xdi + bi1Xdi\u22121 + … + bidi, di > 0, where X is a positive real number and bij\u2019s are integers. Let B = max({|bij|}i,j,1) and d = maxi{di}. If X > (B +1)(n\u00b7d)O(1) then a non-zero S =Pn i=1 \u03b4i \u00b7\u221aai is lower bounded as |S|\u2265 1\/X(n\u00b7d)O(1). The constant in the O(1) notation, as \ufb01xed by our analysis, is roughly 2. We then consider the following more general problem: given an arithmetic circuit computing a multivariate polynomial f(X) and integer d, is the degree of f(X) less than or equal to d? We give a coRPPP-algorithm for this problem, improving previous results of Allender, Bu\u00a8rgisser, Kjeldgaard-Pedersen and Miltersen (2009), and Koiran and Perifel (2007).<\/p>\n","protected":false},"excerpt":{"rendered":"
The sum of square roots problem over integers is the task of deciding the sign of a non-zero sum, S =Pn i=1 \u03b4i \u00b7\u221aai, where \u03b4i \u2208 {+1,\u22121} and ai\u2019s are positive integers that are upperbounded by N (say). A fundamental open question in numerical analysis and computational geometry is whether |S|\u22651\/2(n\u00b7log N)O(1) when S […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":[{"type":"user_nicename","value":"neeraka"},{"type":"text","value":"Chandan Saha"}],"msr_publishername":"IEEE","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"Conference on Computational Complexity (CCC)","msr_doi":"","msr_arxiv_id":"","msr_mag_id":"","msr_other_authors":"Chandan 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