compute a rational univariate representation
RationalUnivariateRepresentation(J, v, opts)
a list or set of polynomials or a PolynomialIdeal
(optional) new variable
optional arguments of the form keyword=value
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.
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.
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.
RationalUnivariateRepresentation does not currently support algebraic extensions (specified by RootOfs or radicals), parameters, or characteristics other than zero.
F ≔ 5⁢x3−330⁢x⁢y+17,3⁢x2⁢y−20⁢y2+x−2
f,d,N ≔ RationalUnivariateRepresentation⁡F,v,output=polynomials
f,d,N ≔ 445⁢v6+12233⁢v3−21780⁢v2−578,8900⁢v5+122330⁢v2−145200⁢v,y=−1395⁢v4+4400⁢v3+6477⁢v−7480,x=−122330⁢v3+290400⁢v2+11560
J ≔ F
J ≔ 5⁢x3−330⁢x⁢y+17,3⁢x2⁢y−20⁢y2+x−2
An example where the univariate polynomial factors:
F ≔ x2+y2−25,x−72+y−72−25
A similar system with a single solution of multiplicity two:
F ≔ x2+y2−25,x−62+y−82−25
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