Hoenders and Slump describe a method for determining number and multiplicities of zeros of a function based on a numerical quadrature technique, but as they show in their Tables 4-5, this technique is unstable. It seems unlikely to us that any numerical technique could decisively solve this problem. Symbolic methods can come to our aid in some instances. First, if is a rational function over a suitable computable extension of , then square-free factorization is applicable. Second, if where is a purely transcendental extension of a computable extension of , is a multiple zero, and where is a rational function, then
are rational functions in and identical to zero. Since is a common root of both and , the resultant
must be zero. Hence is a root of and is algebraic over . The equation can be solved symbolically for and then this solution substituted into and to ascertain if proving is a multiple zero. Third, if can be decomposed as a composition , is a root of with multiplicity , is a root of with multiplicity , then is a root of with multiplicity .
A piecewise holomorphic function defined by
consisting of finitely many holomorphic may be treated as separate inputs to our algorithm.
An expression containing non-holomorphic subexpressions can sometimes be rewritten to become holomorphic or piecewise holomorphic. As an example, suppose where and are holomorphic. Then solve and replace by better expressions .
Our algorithm is restricted to a finite interval . In some cases, such as , this is necessary, for otherwise there would be an infinite number of solutions. In other cases, such as , there are only finitely many solutions even on . The strategy proposed in this case is to use some asymptotic analysis to find such that is non-zero on and leaving only to contend with.
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