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Computing Rational Radical Sums in Uniform TC^0

Authors Paul Hunter, Patricia Bouyer, Nicolas Markey, Joël Ouaknine, James Worrell

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LIPIcs.FSTTCS.2010.308.pdf
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  • Sum of square roots
  • Threshold circuits
  • Complexity

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Abstract

A fundamental problem in numerical computation and computational geometry is to determine the sign of arithmetic expressions in radicals. Here we consider the simpler problem of deciding whether $\sum_{i=1}^m C_i A_i^{X_i}$ is zero for given rational numbers $A_i$, $C_i$, $X_i$. It has been known for almost twenty years that this can be decided in polynomial time. In this paper we improve this result by showing membership in uniform TC0. This requires several significant departures from Blömer's polynomial-time algorithm as the latter crucially relies on primitives, such as gcd computation and binary search, that are not known to be in TC0.

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Paul Hunter, Patricia Bouyer, Nicolas Markey, Joël Ouaknine, and James Worrell. Computing Rational Radical Sums in Uniform TC^0. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 308-316, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010) https://doi.org/10.4230/LIPIcs.FSTTCS.2010.308

Author Details

Paul Hunter
Patricia Bouyer
Nicolas Markey
Joël Ouaknine
James Worrell
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