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Groebner

 RationalUnivariateRepresentation
 compute a rational univariate representation

 Calling Sequence RationalUnivariateRepresentation(J, v, opts)

Parameters

 J - a list or set of polynomials or a PolynomialIdeal v - (optional) new variable opts - optional arguments of the form keyword=value

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Groebner}\right):$
 > $F≔\left[5{x}^{3}-330xy+17,3{x}^{2}y-20{y}^{2}+x-2\right]$
 ${F}{≔}\left[{5}{}{{x}}^{{3}}{-}{330}{}{x}{}{y}{+}{17}{,}{3}{}{{x}}^{{2}}{}{y}{-}{20}{}{{y}}^{{2}}{+}{x}{-}{2}\right]$ (1)
 > $\mathrm{IsZeroDimensional}\left(F\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{Groebner}\left[\mathrm{Basis}\right]\left(F,\mathrm{plex}\left(x,y\right)\right)$
 $\left[{15842000}{}{{y}}^{{6}}{+}{1228200}{}{{y}}^{{4}}{-}{75993}{}{{y}}^{{3}}{-}{33600}{}{{y}}^{{2}}{-}{1770}{}{y}{+}{285}{,}{133500534000}{}{{y}}^{{5}}{+}{2386755720000}{}{{y}}^{{4}}{+}{35538821400}{}{{y}}^{{3}}{+}{211467699989}{}{{y}}^{{2}}{+}{1260279815}{}{x}{-}{5026814580}{}{y}{-}{2748131560}\right]$ (3)
 > $\mathrm{RationalUnivariateRepresentation}\left(F,v\right)$
 ${445}{}{{v}}^{{6}}{+}{12233}{}{{v}}^{{3}}{-}{21780}{}{{v}}^{{2}}{-}{578}{=}{0}{,}\left\{{x}{=}\frac{{-}{122330}{}{{v}}^{{3}}{+}{290400}{}{{v}}^{{2}}{+}{11560}}{{8900}{}{{v}}^{{5}}{+}{122330}{}{{v}}^{{2}}{-}{145200}{}{v}}{,}{y}{=}\frac{{-}{1395}{}{{v}}^{{4}}{+}{4400}{}{{v}}^{{3}}{+}{6477}{}{v}{-}{7480}}{{8900}{}{{v}}^{{5}}{+}{122330}{}{{v}}^{{2}}{-}{145200}{}{v}}\right\}$ (4)
 > $f,d,N≔\mathrm{RationalUnivariateRepresentation}\left(F,v,\mathrm{output}=\mathrm{polynomials}\right)$
 ${f}{,}{d}{,}{N}{≔}{445}{}{{v}}^{{6}}{+}{12233}{}{{v}}^{{3}}{-}{21780}{}{{v}}^{{2}}{-}{578}{,}{8900}{}{{v}}^{{5}}{+}{122330}{}{{v}}^{{2}}{-}{145200}{}{v}{,}\left[{y}{=}{-}{1395}{}{{v}}^{{4}}{+}{4400}{}{{v}}^{{3}}{+}{6477}{}{v}{-}{7480}{,}{x}{=}{-}{122330}{}{{v}}^{{3}}{+}{290400}{}{{v}}^{{2}}{+}{11560}\right]$ (5)
 > $\mathrm{factor}\left(f\right)$
 ${445}{}{{v}}^{{6}}{+}{12233}{}{{v}}^{{3}}{-}{21780}{}{{v}}^{{2}}{-}{578}$ (6)
 > $\mathrm{with}\left(\mathrm{PolynomialIdeals}\right):$
 > $J≔⟨F⟩$
 ${J}{≔}⟨{5}{}{{x}}^{{3}}{-}{330}{}{x}{}{y}{+}{17}{,}{3}{}{{x}}^{{2}}{}{y}{-}{20}{}{{y}}^{{2}}{+}{x}{-}{2}⟩$ (7)
 > $\mathrm{IsPrime}\left(J\right)$
 ${\mathrm{true}}$ (8)

An example where the univariate polynomial factors:

 > $F≔\left[{x}^{2}+{y}^{2}-25,{\left(x-7\right)}^{2}+{\left(y-7\right)}^{2}-25\right]$
 ${F}{≔}\left[{{x}}^{{2}}{+}{{y}}^{{2}}{-}{25}{,}{\left({x}{-}{7}\right)}^{{2}}{+}{\left({y}{-}{7}\right)}^{{2}}{-}{25}\right]$ (9)
 > $\mathrm{RationalUnivariateRepresentation}\left(F,v\right)$
 ${{v}}^{{2}}{-}{7}{}{v}{+}{12}{=}{0}{,}\left\{{x}{=}\frac{{-}{24}{+}{7}{}{v}}{{-}{7}{+}{2}{}{v}}{,}{y}{=}\frac{{-}{25}{+}{7}{}{v}}{{-}{7}{+}{2}{}{v}}\right\}$ (10)
 > $\mathrm{RationalUnivariateRepresentation}\left(F,v,\mathrm{output}=\mathrm{factored}\right)$
 $\left[{v}{-}{3}{,}\left[{y}{=}{4}{,}{x}{=}{3}\right]\right]{,}\left[{v}{-}{4}{,}\left[{y}{=}{3}{,}{x}{=}{4}\right]\right]$ (11)

A similar system with a single solution of multiplicity two:

 > $F≔\left[{x}^{2}+{y}^{2}-25,{\left(x-6\right)}^{2}+{\left(y-8\right)}^{2}-25\right]$
 ${F}{≔}\left[{{x}}^{{2}}{+}{{y}}^{{2}}{-}{25}{,}{\left({x}{-}{6}\right)}^{{2}}{+}{\left({y}{-}{8}\right)}^{{2}}{-}{25}\right]$ (12)
 > $\mathrm{RationalUnivariateRepresentation}\left(F,v\right)$
 ${{v}}^{{2}}{-}{6}{}{v}{+}{9}{=}{0}{,}\left\{{x}{=}{v}{,}{y}{=}{4}\right\}$ (13)
 > $\mathrm{RationalUnivariateRepresentation}\left(F,v,\mathrm{output}=\mathrm{factored}\right)$
 $\left[{v}{-}{3}{,}\left[{y}{=}{4}{,}{x}{=}{3}\right]\right]{,}\left[{v}{-}{3}{,}\left[{y}{=}{4}{,}{x}{=}{3}\right]\right]$ (14)

 See Also