RationalUnivariateRepresentation - Maple Help
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Groebner

  

RationalUnivariateRepresentation

  

compute a rational univariate representation

 

Calling Sequence

Parameters

Description

Examples

References

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

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

An example where the univariate polynomial factors:

(9)

(10)

(11)

A similar system with a single solution of multiplicity two:

(12)

(13)

(14)

References

  

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.

See Also

Basis

FGLM

IsPrime

IsZeroDimensional

 


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