Recognising zero among implicitly defined elementary numbers

It would be desirable to have an algorithm to decide equality among the constants which commonly occur in scientific computing. We do not yet know whether or not this is possible. It is known, however, that if the Schanuel conjecture is true, then equality is Turing decidable among the elementary numbers, that is, the complex numbers which can be defined by systems of equations built up from a list of variables and the rational numbers using field operations, and the exponential function. An algorithm based on the Schanuel conjecture is described in this article to decide equality among these numbers. The algebraic part of this algorithm is simpler than those which have been given previously to solve related problems, depending essentially on the zero problem for implicitly defined algebraic numbers.

[1]  Chee-Keng Yap,et al.  Recent progress in exact geometric computation , 2005, J. Log. Algebraic Methods Program..

[2]  Chen Li,et al.  Exact Geometric Computation: Theory and Applications , 2001 .

[3]  Daniel Richardson,et al.  Zero Tests for Constants in Simple Scientific Computation , 2007, Math. Comput. Sci..

[4]  Chee-Keng Yap,et al.  A new constructive root bound for algebraic expressions , 2001, SODA '01.

[5]  Jihun Yu,et al.  Foundations of Exact Rounding , 2009, WALCOM.

[6]  A. Neumaier.  Interval methods for systems of equations , 1990 .

[7]  B. F. Caviness,et al.  A note on algebraic independence of logarithmic and exponential constants , 1978, SIGS.

[8]  G. Alefeld,et al.  Introduction to Interval Computation , 1983 .

[9]  Chee-Keng Yap,et al.  Robust Geometric Computation , 2016, Encyclopedia of Algorithms.

[10]  Kurt Mehlhorn,et al.  Exact geometric computation made easy , 1999 .

[11]  John Canny,et al.  The complexity of robot motion planning , 1988 .

[12]  Chee-Keng Yap,et al.  A core library for robust numeric and geometric computation , 1999, SCG '99.

[13]  Chee-Keng Yap,et al.  Fundamental problems of algorithmic algebra , 1999 .

[14]  Chee Yap.  On guaranteed accuracy computation , 2004 .

[15]  Angus Macintyre,et al.  On the decidability of the real exponential field , 1996 .

[16]  John Shackell.  Zero-equivalence in function fields defined by algebraic differential equations , 1993 .

[17]  Adam W. Strzebonski.  Real root isolation for exp-log functions , 2008, ISSAC '08.