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2012 IEEE 27th Conference on Computational Complexity (2011)
San Jose, California USA
June 8, 2011 to June 11, 2011
ISSN: 1093-0159
ISBN: 978-0-7695-4411-3
pp: 292-299
The sum of square roots problem over integers is the task of deciding the sign of a non-zero sum, S = \sum_{i=1}^{n}{\delta_i \cdot \sqrt{a_i}}, where \delta_i \in \{ +1, -1\}$ and $a_i's are positive integers that are upper bounded by N (say). A fundamental open question in numerical analysis and computational geometry is whether \abs{S} \geq 1/2^{(n \cdot \log N)^{O(1)}} when S \neq 0. We study a formulation of this problem over polynomials: Given an expression S = \sum_{i=1}^{n}{c_i \cdot \sqrt{f_i(x)}}, where c_i's belong to a field of characteristic 0 and f_i's are univariate polynomials with degree bounded by d and f_i(0) \neq 0$ for all $i, is it true that the minimum exponent of x which has a nonzero coefficient in the power series S is upper bounded by (n \cdot d)^{O(1)}, unless S=0$? 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 a_i is of the form, a_i = X^{d_i} + b_{i 1} X^{d_i - 1} + \ldots + b_{i d_i, \hspace{0.1in} d_i > 0, , where X is a positive real number and b_{i j}'s are integers. Let B = \max_{i,j}\{\abs{b_{i j}}\} and d = \max_i\{d_i\}$. If $X > (B+1)^{(n \cdot d)^{O(1)}} then a \emph{non-zero} S = \sum_{i=1}^{n}{\delta_i \cdot \sqrt{a_i}} is lower bounded as \abs{S} \geq 1/X^{(n \cdot d)^{O(1)}}. The constant in the O(1) notation, 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(\vecx) and integer d, is the degree of f(\vecx) less than or equal to d? We give a , \mathsf{coRP}^{\mathsf{PP}}-algorithm for this problem, improving previous results of \cite{ABKM09} and \cite{KP07}.
Sum of square roots, arithmetic circuit complexity
Chandan Saha, Neeraj Kayal, "On the Sum of Square Roots of Polynomials and Related Problems", 2012 IEEE 27th Conference on Computational Complexity, vol. 00, no. , pp. 292-299, 2011, doi:10.1109/CCC.2011.19
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