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On the Sum of Square Roots of Polynomials and Related Problems

Published:01 November 2012Publication History
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Abstract

The sum of square roots over integers problem is the task of deciding the sign of a nonzero sum, S = ∑i=1n δi · √ai, where δi ∈ {+1, −1} and ai’s are positive integers that are upper bounded by N (say). A fundamental open question in numerical analysis and computational geometry is whether ∣S∣ ≥ 1/2(n ·log N)O(1) when S ≠ 0. We study a formulation of this problem over polynomials. Given an expression S = ∑i=1n ci · √fi(x), where ci’s belong to a field of characteristic 0 and fi’s are univariate polynomials with degree bounded by d and fi(0)≠0 for all i, is it true that the minimum exponent of x that has a nonzero coefficient in the power series S is upper bounded by (n · d)O(1), unless S = 0? We answer this question affirmatively. 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−1 +...+ bidi, di > 0, where X is a positive real number and bij’s are integers. Let B = max ({∣bij∣}i, j, 1) and d = maxi{di}. If X > (B + 1)(n·d)O(1) then a nonzero S = ∑i=1n δi · √ai is lower bounded as ∣S∣ ≥ 1/X(n·d)O(1). The constant in O(1), as fixed 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 et al. [2009] and Koiran and Perifel [2007].

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      • Published in

        cover image ACM Transactions on Computation Theory
        ACM Transactions on Computation Theory  Volume 4, Issue 4
        November 2012
        57 pages
        ISSN:1942-3454
        EISSN:1942-3462
        DOI:10.1145/2382559
        Issue’s Table of Contents

        Copyright © 2012 ACM

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 1 November 2012
        • Accepted: 1 June 2012
        • Revised: 1 March 2012
        • Received: 1 September 2011
        Published in toct Volume 4, Issue 4

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