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].
- Allender, E., Bürgisser, P., Kjeldgaard-Pedersen, J., and Miltersen, P. B. 2009. On the complexity of numerical analysis. SIAM J. Comput. 38, 5, 1987--2006. Google Scholar
Digital Library
- Blömer, J. 1991. Computing sums of radicals in polynomial time. In Proceedings of the 32nd IEEE Symposium on Foundations of Computer Science (FOCS). 670--677. Google Scholar
Digital Library
- Blömer, J. 1993. Computing sums of radicals in polynomial time. http://www.cs.uni-paderborn.de/uploads/tx_sibibtex/ComputingSumsOfRadicals.pdf.Google Scholar
- Bôcher, M. 1900. On linear dependence of functions of one variables. Bull. Amer. Math. Soc. 7, 120--121.Google Scholar
Cross Ref
- Bostan, A. and Dumas, P. 2010. Wronskians and linear independence. Amer. Math. Monthly 117, 8, 722--727.Google Scholar
Cross Ref
- Brent, R. P. 1976. Fast multiple-precision evaluation of elementary functions. J. ACM 23, 2, 242--251. Google Scholar
Digital Library
- Burnikel, C., Fleischer, R., Mehlhorn, K., and Schirra, S. 2000. A strong and easily computable separation bound for arithmetic expressions involving radicals. Algorithmica 27, 1, 87--99.Google Scholar
Cross Ref
- Cheng, Q. 2006. On comparing sums of square roots of small integers. In Proceedings of MFCS. 250--255. Google Scholar
Digital Library
- Cheng, Q., Meng, X., Sun, C., and Chen, J. 2010. Bounding the sum of square roots via lattice reduction. Math. Computat. 79, 1109--1122.Google Scholar
Cross Ref
- Effinger, G., Hicks, K. H., and Mullen, G. L. 2005. Integers and Polynomials: Comparing the close cousins Z and F{x}. Math. Intell. 27, 26--34.Google Scholar
Cross Ref
- Garey, M. R., Graham, R. L., and Johnson, D. S. 1976. Some NP-Complete Geometric Problems. In Proceedings of STOC. 10--22. Google Scholar
Digital Library
- Goemans, M. 1998. Semidefinite programming and combinatorial optimization. Documenta Mathematica, Extra Volume, Proceedings ICM III, 657--666.Google Scholar
- Immerman, N. and Landau, S. 1993. The similarities (and differences) between polynomials and integers. In Proceedings International Conference on Number Theoretic and Algebraic Methods in Computer Science. 57--59.Google Scholar
- Koiran, P. and Perifel, S. 2007. The complexity of two problems on arithmetic circuits. Theor. Comput. Sci. 389, 1--2, 172--181. Google Scholar
Digital Library
- Malojovich, G. 1996. An effective version of Kronecker’s theorem on simultaneous Diophantine equation. Tech. rep., City University of Hong Kong.Google Scholar
- Mehlhorn, K. and Schirra, S. 2000. Generalized and improved constructive separation bound for real algebraic expressions. Resear. rep. MPI-I-2000-1-004, Max-Planck-Institut für Informatik, Saarbrücken, Germany.Google Scholar
- Mulzer, W. and Rote, G. 2008. Minimum-weight triangulation is NP-hard. J. ACM 55, 2. Google Scholar
Digital Library
- Qian, J. and Wang, C. A. 2006. How much precision is needed to compare two sums of square roots of integers? Inf. Process. Lett. 100, 5, 194--198. Google Scholar
Digital Library
- Reif, J. H. and Tate, S. R. 1992. On threshold circuits and polynomial computation. SIAM J. Comput. 21, 5, 896--908. Google Scholar
Digital Library
- Tiwari, P. 1992. A problem that is easier to solve on the unit-cost algebraic RAM. J. Complexity 8, 393--397. Google Scholar
Digital Library
- van Lint, J. H. 1999. Introduction to Coding Theory. Springer. Google Scholar
Digital Library
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On the Sum of Square Roots of Polynomials and Related Problems
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