Editing Risch algorithm (section)

==Decidability==
The Risch algorithm applied to general elementary functions is not an algorithm but a [[RE (complexity)|semi-algorithm]] because it needs to check, as a part of its operation, if certain expressions are equivalent to zero ([[constant problem]]), in particular in the constant field. For expressions that involve only functions commonly taken to be [[elementary function|elementary]] it is not known whether an algorithm performing such a check exists (current [[computer algebra system]]s use heuristics); moreover, if one adds the [[absolute value|absolute value function]] to the list of elementary functions, then it is known that no such algorithm exists; see [[Richardson's theorem]].

This issue also arises in the [[polynomial division algorithm]]; this algorithm will fail if it cannot correctly determine whether coefficients vanish identically.<ref>{{Cite web| title= Mathematica 7 Documentation: PolynomialQuotient| url= http://reference.wolfram.com/mathematica/ref/PolynomialQuotient.html| work= Section: Possible Issues| access-date= July 17, 2010}}</ref> Virtually every non-trivial algorithm relating to polynomials uses the polynomial division algorithm, the Risch algorithm included.  If the constant field is [[computable]], i.e., for elements not dependent on {{math|''x''}}, then the problem of zero-equivalence is decidable, so the Risch algorithm is a complete algorithm. Examples of computable constant fields are {{math|&Qopf;}} and {{math|&Qopf;(''y'')}}, i.e., rational numbers and rational functions in {{mvar|''y''}} with rational-number coefficients, respectively, where {{math|''y''}} is an indeterminate that does not depend on {{math|''x''}}.

This is also an issue in the [[Gaussian elimination]] matrix algorithm (or any algorithm that can compute the [[nullspace]] of a matrix), which is also necessary for many parts of the Risch algorithm.  Gaussian elimination will produce incorrect results if it cannot correctly determine whether a pivot is identically zero{{Citation needed|date=January 2012}}.