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Groebner

  

RationalUnivariateRepresentation

  

compute a rational univariate representation

 

Calling Sequence

Parameters

Description

Examples

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

withGroebner:

F5x3330xy+17,3x2y20y2+x2

F:=5x3330xy+17,3x2y20y2+x2

(1)

IsZeroDimensionalF

true

(2)

Groebner[Basis]F,plexx,y

15842000y6+1228200y475993y333600y21770y+285,133500534000y5+2386755720000y4+35538821400y3+211467699989y2+1260279815x5026814580y2748131560

(3)

RationalUnivariateRepresentationF,v

445v6+12233v321780v2578=0,x=122330v3+290400v2+115608900v5+122330v2145200v,y=1395v4+4400v3+6477v74808900v5+122330v2145200v

(4)

f,d,NRationalUnivariateRepresentationF,v,output=polynomials

f,d,N:=445v6+12233v321780v2578,8900v5+122330v2145200v,y=1395v4+4400v3+6477v7480,x=122330v3+290400v2+11560

(5)

factorf

445v6+12233v321780v2578

(6)

withPolynomialIdeals:

JF

J:=5x3330xy+17,3x2y20y2+x2

(7)

IsPrimeJ

true

(8)

An example where the univariate polynomial factors:

Fx2+y225,x72+y7225

F:=x2+y225,x72+y7225

(9)

RationalUnivariateRepresentationF,v

v27v+12=0,x=24+7v7+2v,y=25+7v7+2v

(10)

RationalUnivariateRepresentationF,v,output=factored

v3,y=4,x=3,v4,y=3,x=4

(11)

A similar system with a single solution of multiplicity two:

Fx2+y225,x62+y8225

F:=x2+y225,x62+y8225

(12)

RationalUnivariateRepresentationF,v

v26v+9=0,x=v,y=4

(13)

RationalUnivariateRepresentationF,v,output=factored

v3,y=4,x=3,v3,y=4,x=3

(14)

See Also

Basis

FGLM

IsPrime

IsZeroDimensional

 


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