Nicolas Markey

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 ∑i=1m Ci . AiXi is zero for given rational numbers Ai, Ci, Xi. 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.

@inproceedings{fsttcs2010-HBMOW,
  author =              {Hunter, Paul and Bouyer, Patricia and Markey,
                         Nicolas and Ouaknine, Jo{\"e}l and Worrell, James},
  title =               {Computing Rational Radical Sums in Uniform
                         {TC\textsuperscript{0}}},
  editor =              {Lodaya, Kamal and Mahajan, Meena},
  booktitle =           {{P}roceedings of the 30th {C}onference on
                         {F}oundations of {S}oftware {T}echnology and
                         {T}heoretical {C}omputer {S}cience ({FSTTCS}'10)},
  acronym =             {{FSTTCS}'10},
  publisher =           {Leibniz-Zentrum f{\"u}r Informatik},
  series =              {Leibniz International Proceedings in Informatics},
  volume =              {8},
  pages =               {308-316},
  year =                {2010},
  month =               dec,
  doi =                 {10.4230/LIPIcs.FSTTCS.2010.308},
  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
                         \(\sum_{i=1}^m C_i \cdot A_i^{X_i}\) 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
                         \(\textsf{TC}^0\). This requires several significant
                         departures from Bl{\"o}mer'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~\(\textsf{TC}^0\).},
}