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.

$\mathrm{with}\left(\mathrm{Groebner}\right)\:$

$F≔\left[5x^3-330xy+17\,3x^{2}y-20y^2+x-2\right]$

$F≔\left[5x^3-330xy+17\,3x^{2}y-20y^2+x-2\right]$

$\mathrm{IsZeroDimensional}\left(F\right)$

$\mathrm{true}$

$\mathrm{Groebner}\left[\mathrm{Basis}\right]\left(F\,\mathrm{plex}\left(x\,y\right)\right)$

$\left[15842000y^6+1228200y^4-75993y^3-33600y^2-1770y+285\,133500534000y^5+2386755720000y^4+35538821400y^3+211467699989y^2+1260279815x-5026814580y-2748131560\right]$

$\mathrm{RationalUnivariateRepresentation}\left(F\,v\right)$

$445v^6+12233v^3-21780v^2-578=0,\left\{x=\frac{-122330v^3+290400v^2+11560}{8900v^5+122330v^2-145200v}\,y=\frac{-1395v^4+4400v^3+6477v-7480}{8900v^5+122330v^2-145200v}\right\}$

$f,d,N≔\mathrm{RationalUnivariateRepresentation}\left(F\,v\,\mathrm{output}=\mathrm{polynomials}\right)$

$f,d,N≔445v^6+12233v^3-21780v^2-578,8900v^5+122330v^2-145200v,\left[y=-1395v^4+4400v^3+6477v-7480\,x=-122330v^3+290400v^2+11560\right]$

$\mathrm{factor}\left(f\right)$

$445v^6+12233v^3-21780v^2-578$

$\mathrm{with}\left(\mathrm{PolynomialIdeals}\right)\:$

$J≔⟨F⟩$

$J≔⟨5x^3-330xy+17\,3x^{2}y-20y^2+x-2⟩$

$\mathrm{IsPrime}\left(J\right)$

$\mathrm{true}$

$F≔\left[x^2+y^2-25\,{\left(x-7\right)}^2+{\left(y-7\right)}^2-25\right]$

$F≔\left[x^2+y^2-25\,{\left(x-7\right)}^2+{\left(y-7\right)}^2-25\right]$

$\mathrm{RationalUnivariateRepresentation}\left(F\,v\right)$

$v^2-7v+12=0,\left\{x=\frac{-24+7v}{-7+2v}\,y=\frac{-25+7v}{-7+2v}\right\}$

$\mathrm{RationalUnivariateRepresentation}\left(F\,v\,\mathrm{output}=\mathrm{factored}\right)$

$\left[v-4\,\left[y=3\,x=4\right]\right],\left[v-3\,\left[y=4\,x=3\right]\right]$

$F≔\left[x^2+y^2-25\,{\left(x-6\right)}^2+{\left(y-8\right)}^2-25\right]$

$F≔\left[x^2+y^2-25\,{\left(x-6\right)}^2+{\left(y-8\right)}^2-25\right]$

$\mathrm{RationalUnivariateRepresentation}\left(F\,v\right)$

$v^2-6v+9=0,\left\{x=v\,y=4\right\}$

$\mathrm{RationalUnivariateRepresentation}\left(F\,v\,\mathrm{output}=\mathrm{factored}\right)$

$\left[v-3\,\left[y=4\,x=3\right]\right],\left[v-3\,\left[y=4\,x=3\right]\right]$

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.