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Calling Sequence
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RationalUnivariateRepresentation(J, v, opts)
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Parameters
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J
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a list or set of polynomials or a PolynomialIdeal
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v
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(optional) new variable
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opts
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optional arguments of the form keyword=value
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Description
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The RationalUnivariateRepresentation command computes a rational univariate representation (or RUR) for a zero-dimensional ideal J. Zero-dimensional systems have a finite number of complex solutions, and an RUR defines a bijection between those solutions and the roots of a univariate polynomial. The advantage of using this representation is that in the worst case the coefficients are an order of magnitude smaller than those of a lexicographic Groebner basis.
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The default output is a sequence consisting of an equation f(v)=0 and a set of substitutions x[i] = u[i](v)/d(v) for each variable x[i]. f(v) is a univariate polynomial defining a common algebraic extension, and the solutions of the system are expressed as rational functions in the new variable v with common denominator d(v). If the v is not specified then the global variable _Z is used by default.
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The optional argument output controls the form of the result. output=polynomials returns the RUR in a format that is more suitable for programming. In this case, the command returns a sequence consisting of f(v), d(v), and a list of x[i] = u[i]. Alternatively, output=factored factors the univariate polynomial f(v) and splits the RUR into a union of multiple reduced RURs in each irreducible component of f(v). The output is returned as a sequence of two-element lists each containing f[j](v) and a list of x[i] = rem(u[i], f[j](v))/rem(d(v), f[j](v)) . Note that the list of factors f[j](v) are not necessarily unique within the output; instead, their multiplicity is preserved. Each factor f[j](v) will also be monic.
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RationalUnivariateRepresentation does not currently support algebraic extensions (specified by RootOfs or radicals), parameters, or characteristics other than zero.
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Examples
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An example where the univariate polynomial factors:
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A similar system with a single solution of multiplicity two:
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References
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Rouillier, F. "Solving zero-dimensional systems through the rational univariate representation." Journal of Applicable Algebra in Engineering, Communication, and Computing, Vol. 9, No. 5 (1999): 433-461.
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