edu.jas.application

Class SolvableIdeal<C extends GcdRingElem<C>>

java.lang.Object

edu.jas.application.SolvableIdeal<C>

All Implemented Interfaces:

java.io.Serializable, java.lang.Comparable<SolvableIdeal<C>>

public class SolvableIdeal<C extends GcdRingElem<C>>
extends java.lang.Object
implements java.lang.Comparable<SolvableIdeal<C>>, java.io.Serializable

Solvable Ideal implements some methods for ideal arithmetic, for example sum, intersection, quotient. Note: only left ideals at the moment.

Author:

Heinz Kredel

See Also:

Serialized Form

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
static class	SolvableIdeal.Side Side variant of ideal.

Field Summary

Fields

Modifier and Type	Field and Description
protected SolvableGroebnerBaseAbstract<C>	bb Groebner base engine.
protected boolean	isGB Indicator if list is a Groebner Base.
protected boolean	isTopt Indicator if list has optimized term order.
protected PolynomialList<C>	list The data structure is a PolynomialList.
protected SolvableReduction<C>	red Reduction engine.
protected SolvableIdeal.Side	sided Indicator of side of Groebner Base.
protected boolean	testGB Indicator if test has been performed if this is a Groebner Base.

Constructor Summary

Constructors

Constructor and Description
SolvableIdeal(GenSolvablePolynomialRing<C> ring) Constructor.
SolvableIdeal(GenSolvablePolynomialRing<C> ring, java.util.List<GenSolvablePolynomial<C>> F) Constructor.
SolvableIdeal(GenSolvablePolynomialRing<C> ring, java.util.List<GenSolvablePolynomial<C>> F, boolean gb) Constructor.
SolvableIdeal(GenSolvablePolynomialRing<C> ring, java.util.List<GenSolvablePolynomial<C>> F, boolean gb, boolean topt) Constructor.
SolvableIdeal(GenSolvablePolynomialRing<C> ring, java.util.List<GenSolvablePolynomial<C>> F, boolean gb, SolvableIdeal.Side s) Constructor.
SolvableIdeal(PolynomialList<C> list) Constructor.
SolvableIdeal(PolynomialList<C> list, boolean gb) Constructor.
SolvableIdeal(PolynomialList<C> list, boolean gb, boolean topt) Constructor.
SolvableIdeal(PolynomialList<C> list, boolean gb, boolean topt, SolvableGroebnerBaseAbstract<C> bb) Constructor.
SolvableIdeal(PolynomialList<C> list, boolean gb, boolean topt, SolvableGroebnerBaseAbstract<C> bb, SolvableReduction<C> red) Constructor.
SolvableIdeal(PolynomialList<C> list, boolean gb, boolean topt, SolvableGroebnerBaseAbstract<C> bb, SolvableReduction<C> red, SolvableIdeal.Side s) Constructor.
SolvableIdeal(PolynomialList<C> list, boolean gb, boolean topt, SolvableIdeal.Side s) Constructor.
SolvableIdeal(PolynomialList<C> list, boolean gb, SolvableGroebnerBaseAbstract<C> bb) Constructor.
SolvableIdeal(PolynomialList<C> list, boolean gb, SolvableGroebnerBaseAbstract<C> bb, SolvableReduction<C> red) Constructor.
SolvableIdeal(PolynomialList<C> list, boolean gb, SolvableIdeal.Side s) Constructor.
SolvableIdeal(PolynomialList<C> list, SolvableGroebnerBaseAbstract<C> bb, SolvableReduction<C> red) Constructor.

Method Summary

Modifier and Type	Method and Description
SolvableIdeal<C>	annihilator(GenSolvablePolynomial<C> h) Annihilator for element modulo this ideal.
SolvableIdeal<C>	annihilator(SolvableIdeal<C> H) Annihilator for ideal modulo this ideal.
int	commonZeroTest() Ideal common zero test.
int	compareTo(SolvableIdeal<C> L) SolvableIdeal comparison.
java.util.List<GenSolvablePolynomial<C>>	constructUnivariate() Construct univariate polynomials of minimal degree in all variables in zero dimensional ideal(G).
GenSolvablePolynomial<C>	constructUnivariate(int i) Construct univariate polynomial of minimal degree in variable i in zero dimensional ideal(G).
boolean	contains(GenSolvablePolynomial<C> b) Solvable ideal containment.
boolean	contains(java.util.List<GenSolvablePolynomial<C>> B) Solvable ideal containment.
boolean	contains(SolvableIdeal<C> B) Solvable ideal containment.
SolvableIdeal<C>	copy() Clone this.
Dimension	dimension() Ideal dimension.
void	doGB() Do Groebner Base. compute the left Groebner Base for this ideal.
SolvableIdeal<C>	eliminate(GenSolvablePolynomialRing<C> R) Eliminate.
boolean	equals(java.lang.Object b) Comparison with any other object.
SolvableIdeal<C>	GB() Groebner Base.
java.util.List<GenSolvablePolynomial<C>>	getList() Get the List of GenSolvablePolynomials.
SolvableIdeal<C>	getONE() Get the one ideal.
GenSolvablePolynomialRing<C>	getRing() Get the GenSolvablePolynomialRing.
SolvableIdeal<C>	getZERO() Get the zero ideal.
int	hashCode() Hash code for this solvable ideal.
SolvableIdeal<C>	infiniteQuotient(GenSolvablePolynomial<C> h) Infinite quotient.
SolvableIdeal<C>	infiniteQuotient(SolvableIdeal<C> H) Infinite Quotient.
int	infiniteQuotientExponent(GenSolvablePolynomial<C> h, SolvableIdeal<C> Q) Infinite quotient exponent.
SolvableIdeal<C>	infiniteQuotientRab(GenSolvablePolynomial<C> h) Infinite quotient.
SolvableIdeal<C>	infiniteQuotientRab(SolvableIdeal<C> H) Infinite Quotient.
SolvableIdeal<C>	intersect(GenSolvablePolynomialRing<C> R) Intersection.
SolvableIdeal<C>	intersect(java.util.List<SolvableIdeal<C>> Bl) Intersection.
SolvableIdeal<C>	intersect(SolvableIdeal<C> B) Intersection.
GenSolvablePolynomial<C>	inverse(GenSolvablePolynomial<C> h) Inverse for element modulo this ideal.
boolean	isAnnihilator(GenSolvablePolynomial<C> h, SolvableIdeal<C> A) Test for annihilator of element modulo this ideal.
boolean	isAnnihilator(SolvableIdeal<C> H, SolvableIdeal<C> A) Test for annihilator of ideal modulo this ideal.
boolean	isGB() Test if this is a left Groebner base.
boolean	isMaximal() Test if this ideal is maximal.
boolean	isONE() Test if ONE is contained in the ideal.
boolean	isRadicalMember(GenSolvablePolynomial<C> h) Radical membership test.
boolean	isRightGB() Test if this is a right Groebner base.
boolean	isTwosidedGB() Test if this is a twosided Groebner base.
boolean	isUnit(GenSolvablePolynomial<C> h) Test if element is a unit modulo this ideal.
boolean	isZERO() Test if ZERO ideal.
GenSolvablePolynomial<C>	normalform(GenSolvablePolynomial<C> h) Normalform for element.
java.util.List<GenSolvablePolynomial<C>>	normalform(java.util.List<GenSolvablePolynomial<C>> L) Normalform for list of solvable elements.
SolvableIdeal<C>	power(int d) Power.
SolvableIdeal<C>	product(GenSolvablePolynomial<C> b) Left product.
SolvableIdeal<C>	product(SolvableIdeal<C> B) Product.
SolvableIdeal<C>	quotient(GenSolvablePolynomial<C> h) Quotient.
SolvableIdeal<C>	quotient(SolvableIdeal<C> H) Quotient.
SolvableIdeal<C>	rightGB() Groebner Base.
SolvableIdeal<C>	sum(GenSolvablePolynomial<C> b) Solvable summation.
SolvableIdeal<C>	sum(java.util.List<GenSolvablePolynomial<C>> L) Solvable summation.
SolvableIdeal<C>	sum(SolvableIdeal<C> B) Solvable ideal summation.
java.lang.String	toScript() Get a scripting compatible string representation.
java.lang.String	toString() String representation of the solvable ideal.
SolvableIdeal<C>	twosidedGB() Groebner Base.
java.util.List<java.lang.Long>	univariateDegrees() Univariate head term degrees.

Methods inherited from class java.lang.Object

clone, finalize, getClass, notify, notifyAll, wait, wait, wait

Field Detail

list

protected PolynomialList<C extends GcdRingElem<C>> list

The data structure is a PolynomialList.

isGB

protected boolean isGB

Indicator if list is a Groebner Base.

sided

protected SolvableIdeal.Side sided

Indicator of side of Groebner Base.

testGB

protected boolean testGB

Indicator if test has been performed if this is a Groebner Base.

isTopt

protected boolean isTopt

Indicator if list has optimized term order.

bb

protected final SolvableGroebnerBaseAbstract<C extends GcdRingElem<C>> bb

Groebner base engine.

red

protected final SolvableReduction<C extends GcdRingElem<C>> red

Reduction engine.

Constructor Detail

SolvableIdeal

public SolvableIdeal(GenSolvablePolynomialRing<C> ring)

Constructor.

Parameters:

ring - solvable polynomial ring

SolvableIdeal

public SolvableIdeal(GenSolvablePolynomialRing<C> ring,
                     java.util.List<GenSolvablePolynomial<C>> F)

Constructor.

Parameters:

ring - solvable polynomial ring

F - list of solvable polynomials

SolvableIdeal

public SolvableIdeal(GenSolvablePolynomialRing<C> ring,
                     java.util.List<GenSolvablePolynomial<C>> F,
                     boolean gb)

Constructor.

Parameters:

ring - solvable polynomial ring

F - list of solvable polynomials

gb - true if F is known to be a Groebner Base, else false

SolvableIdeal

public SolvableIdeal(GenSolvablePolynomialRing<C> ring,
                     java.util.List<GenSolvablePolynomial<C>> F,
                     boolean gb,
                     boolean topt)

Constructor.

Parameters:

ring - solvable polynomial ring

F - list of solvable polynomials

gb - true if F is known to be a Groebner Base, else false

topt - true if term order is optimized, else false

SolvableIdeal

public SolvableIdeal(GenSolvablePolynomialRing<C> ring,
                     java.util.List<GenSolvablePolynomial<C>> F,
                     boolean gb,
                     SolvableIdeal.Side s)

Constructor.

Parameters:

ring - solvable polynomial ring

F - list of solvable polynomials

gb - true if F is known to be a Groebner Base, else false

s - side variant of ideal or Groebner Base

SolvableIdeal

public SolvableIdeal(PolynomialList<C> list)

Constructor.

Parameters:

list - solvable polynomial list

SolvableIdeal

public SolvableIdeal(PolynomialList<C> list,
                     SolvableGroebnerBaseAbstract<C> bb,
                     SolvableReduction<C> red)

Constructor.

Parameters:

list - solvable polynomial list

bb - Groebner Base engine

red - Reduction engine

SolvableIdeal

public SolvableIdeal(PolynomialList<C> list,
                     boolean gb)

Constructor.

Parameters:

list - solvable polynomial list

gb - true if list is known to be a Groebner Base, else false

SolvableIdeal

public SolvableIdeal(PolynomialList<C> list,
                     boolean gb,
                     boolean topt)

Constructor.

Parameters:

list - solvable polynomial list

gb - true if list is known to be a Groebner Base, else false

topt - true if term order is optimized, else false

SolvableIdeal

public SolvableIdeal(PolynomialList<C> list,
                     boolean gb,
                     SolvableIdeal.Side s)

Constructor.

Parameters:

list - solvable polynomial list

gb - true if list is known to be a Groebner Base, else false

s - side variant of ideal or Groebner Base

SolvableIdeal

public SolvableIdeal(PolynomialList<C> list,
                     boolean gb,
                     boolean topt,
                     SolvableIdeal.Side s)

Constructor.

Parameters:

list - solvable polynomial list

gb - true if list is known to be a Groebner Base, else false

topt - true if term order is optimized, else false

s - side variant of ideal or Groebner Base

SolvableIdeal

public SolvableIdeal(PolynomialList<C> list,
                     boolean gb,
                     SolvableGroebnerBaseAbstract<C> bb,
                     SolvableReduction<C> red)

Constructor.

Parameters:

list - solvable polynomial list

gb - true if list is known to be a Groebner Base, else false

bb - Groebner Base engine

red - Reduction engine

SolvableIdeal

public SolvableIdeal(PolynomialList<C> list,
                     boolean gb,
                     SolvableGroebnerBaseAbstract<C> bb)

Constructor.

Parameters:

list - solvable polynomial list

gb - true if list is known to be a Groebner Base, else false

bb - Groebner Base engine

SolvableIdeal

public SolvableIdeal(PolynomialList<C> list,
                     boolean gb,
                     boolean topt,
                     SolvableGroebnerBaseAbstract<C> bb)

Constructor.

Parameters:

list - solvable polynomial list

gb - true if list is known to be a Groebner Base, else false

topt - true if term order is optimized, else false

bb - Groebner Base engine

SolvableIdeal

public SolvableIdeal(PolynomialList<C> list,
                     boolean gb,
                     boolean topt,
                     SolvableGroebnerBaseAbstract<C> bb,
                     SolvableReduction<C> red)

Constructor.

Parameters:

list - solvable polynomial list

gb - true if list is known to be a Groebner Base, else false

topt - true if term order is optimized, else false

bb - Groebner Base engine

red - Reduction engine

SolvableIdeal

public SolvableIdeal(PolynomialList<C> list,
                     boolean gb,
                     boolean topt,
                     SolvableGroebnerBaseAbstract<C> bb,
                     SolvableReduction<C> red,
                     SolvableIdeal.Side s)

Constructor.

Parameters:

list - solvable polynomial list

gb - true if list is known to be a Groebner Base, else false

topt - true if term order is optimized, else false

bb - Groebner Base engine

red - Reduction engine

s - side variant of ideal or Groebner Base

Method Detail

copy

public SolvableIdeal<C> copy()

Clone this.

Returns:

a copy of this.

getList

public java.util.List<GenSolvablePolynomial<C>> getList()

Get the List of GenSolvablePolynomials.

Returns:

(cast) list.list

getRing

public GenSolvablePolynomialRing<C> getRing()

Get the GenSolvablePolynomialRing.

Returns:

(cast) list.ring

getZERO

public SolvableIdeal<C> getZERO()

Get the zero ideal.

Returns:

ideal(0)

getONE

public SolvableIdeal<C> getONE()

Get the one ideal.

Returns:

ideal(1)

toString

public java.lang.String toString()

String representation of the solvable ideal.

Overrides:

toString in class java.lang.Object

See Also:

Object.toString()

toScript

public java.lang.String toScript()

Get a scripting compatible string representation.

Returns:

script compatible representation for this Element.

See Also:

Element.toScript()

equals

public boolean equals(java.lang.Object b)

Comparison with any other object. Note: If not both ideals are Groebner Bases, then false may be returned even the ideals are equal.

Overrides:

equals in class java.lang.Object

See Also:

Object.equals(java.lang.Object)

compareTo

public int compareTo(SolvableIdeal<C> L)

SolvableIdeal comparison.

Specified by:

compareTo in interface java.lang.Comparable<SolvableIdeal<C extends GcdRingElem<C>>>

Parameters:

L - other solvable ideal.

Returns:

compareTo() of polynomial lists.

hashCode

public int hashCode()

Hash code for this solvable ideal.

Overrides:

hashCode in class java.lang.Object

See Also:

Object.hashCode()

isZERO

public boolean isZERO()

Test if ZERO ideal.

Returns:

true, if this is the 0 ideal, else false

isONE

public boolean isONE()

Test if ONE is contained in the ideal. To test for a proper ideal use ! id.isONE().

Returns:

true, if this is the 1 ideal, else false

isGB

public boolean isGB()

Test if this is a left Groebner base.

Returns:

true, if this is a left Groebner base, else false

doGB

public void doGB()

Do Groebner Base. compute the left Groebner Base for this ideal.

GB

public SolvableIdeal<C> GB()

Groebner Base. Get a left Groebner Base for this ideal.

Returns:

leftGB(this)

isTwosidedGB

public boolean isTwosidedGB()

Test if this is a twosided Groebner base.

Returns:

true, if this is a twosided Groebner base, else false

twosidedGB

public SolvableIdeal<C> twosidedGB()

Groebner Base. Get a twosided Groebner Base for this ideal.

Returns:

twosidedGB(this)

isRightGB

public boolean isRightGB()

Test if this is a right Groebner base.

Returns:

true, if this is a right Groebner base, else false

rightGB

public SolvableIdeal<C> rightGB()

Groebner Base. Get a right Groebner Base for this ideal.

Returns:

rightGB(this)

contains

public boolean contains(SolvableIdeal<C> B)

Solvable ideal containment. Test if B is contained in this ideal. Note: this is eventually modified to become a Groebner Base.

Parameters:

B - solvable ideal

Returns:

true, if B is contained in this, else false

contains

public boolean contains(GenSolvablePolynomial<C> b)

Solvable ideal containment. Test if b is contained in this ideal. Note: this is eventually modified to become a Groebner Base.

Parameters:

b - solvable polynomial

Returns:

true, if b is contained in this, else false

contains

public boolean contains(java.util.List<GenSolvablePolynomial<C>> B)

Solvable ideal containment. Test if each b in B is contained in this ideal. Note: this is eventually modified to become a Groebner Base.

Parameters:

B - list of solvable polynomials

Returns:

true, if each b in B is contained in this, else false

sum

public SolvableIdeal<C> sum(SolvableIdeal<C> B)

Solvable ideal summation. Generators for the sum of ideals. Note: if both ideals are Groebner bases, a Groebner base is returned.

Parameters:

B - solvable ideal

Returns:

ideal(this+B)

sum

public SolvableIdeal<C> sum(GenSolvablePolynomial<C> b)

Solvable summation. Generators for the sum of ideal and a polynomial. Note: if this ideal is a Groebner base, a Groebner base is returned.

Parameters:

b - solvable polynomial

Returns:

ideal(this+{b})

sum

public SolvableIdeal<C> sum(java.util.List<GenSolvablePolynomial<C>> L)

Solvable summation. Generators for the sum of this ideal and a list of polynomials. Note: if this ideal is a Groebner base, a Groebner base is returned.

Parameters:

L - list of solvable polynomials

Returns:

ideal(this+L)

product

public SolvableIdeal<C> product(SolvableIdeal<C> B)

Product. Generators for the product of ideals. Note: if both ideals are Groebner bases, a Groebner base is returned.

Parameters:

B - solvable ideal

Returns:

ideal(this*B)

product

public SolvableIdeal<C> product(GenSolvablePolynomial<C> b)

Left product. Generators for the product this by a polynomial.

Parameters:

b - solvable polynomial

Returns:

ideal(this*b)

intersect

public SolvableIdeal<C> intersect(java.util.List<SolvableIdeal<C>> Bl)

Intersection. Generators for the intersection of ideals. Using an iterative algorithm.

Parameters:

Bl - list of solvable ideals

Returns:

ideal(cap_i B_i), a Groebner base

intersect

public SolvableIdeal<C> intersect(SolvableIdeal<C> B)

Intersection. Generators for the intersection of ideals.

Parameters:

B - solvable ideal

Returns:

ideal(this \cap B), a Groebner base

intersect

public SolvableIdeal<C> intersect(GenSolvablePolynomialRing<C> R)

Intersection. Generators for the intersection of a ideal with a polynomial ring. The polynomial ring of this ideal must be a contraction of R and the TermOrder must be an elimination order.

Parameters:

R - solvable polynomial ring

Returns:

ideal(this \cap R)

eliminate

public SolvableIdeal<C> eliminate(GenSolvablePolynomialRing<C> R)

Eliminate. Generators for the intersection of this ideal with a solvable polynomial ring. The solvable polynomial ring of this ideal must be a contraction of R and the TermOrder must be an elimination order.

Parameters:

R - solvable polynomial ring

Returns:

ideal(this \cap R)

quotient

public SolvableIdeal<C> quotient(GenSolvablePolynomial<C> h)

Quotient. Generators for the solvable ideal quotient.

Parameters:

h - solvable polynomial

Returns:

ideal(this : h), a Groebner base

quotient

public SolvableIdeal<C> quotient(SolvableIdeal<C> H)

Quotient. Generators for the solvable ideal quotient.

Parameters:

H - solvable ideal

Returns:

ideal(this : H), a Groebner base

infiniteQuotientRab

public SolvableIdeal<C> infiniteQuotientRab(GenSolvablePolynomial<C> h)

Infinite quotient. Generators for the infinite solvable ideal quotient.

Parameters:

h - solvable polynomial

Returns:

ideal(this : h^s), a Groebner base

infiniteQuotientExponent

public int infiniteQuotientExponent(GenSolvablePolynomial<C> h,
                                    SolvableIdeal<C> Q)

Infinite quotient exponent.

Parameters:

h - solvable polynomial

Q - quotient this : h^\infinity

Returns:

s with Q = this : h^s

infiniteQuotient

public SolvableIdeal<C> infiniteQuotient(GenSolvablePolynomial<C> h)

Infinite quotient. Generators for the infinite solvable ideal quotient.

Parameters:

h - solvable polynomial

Returns:

ideal(this : h^s), a Groebner base

isRadicalMember

public boolean isRadicalMember(GenSolvablePolynomial<C> h)

Radical membership test.

Parameters:

h - solvable polynomial

Returns:

true if h is contained in the radical of ideal(this), else false.

infiniteQuotient

public SolvableIdeal<C> infiniteQuotient(SolvableIdeal<C> H)

Infinite Quotient. Generators for the solvable ideal infinite quotient.

Parameters:

H - solvable ideal

Returns:

ideal(this : H^s), a Groebner base

infiniteQuotientRab

public SolvableIdeal<C> infiniteQuotientRab(SolvableIdeal<C> H)

Infinite Quotient. Generators for the solvable ideal infinite quotient.

Parameters:

H - solvable ideal

Returns:

ideal(this : H^s), a Groebner base

power

public SolvableIdeal<C> power(int d)

Power. Generators for the power of this solvable ideal. Note: if this ideal is a Groebner base, a Groebner base is returned.

Parameters:

d - integer

Returns:

ideal(this^d)

normalform

public GenSolvablePolynomial<C> normalform(GenSolvablePolynomial<C> h)

Normalform for element.

Parameters:

h - solvable polynomial

Returns:

left normalform of h with respect to this

normalform

public java.util.List<GenSolvablePolynomial<C>> normalform(java.util.List<GenSolvablePolynomial<C>> L)

Normalform for list of solvable elements.

Parameters:

L - solvable polynomial list

Returns:

list of left normalforms of the elements of L with respect to this

annihilator

public SolvableIdeal<C> annihilator(GenSolvablePolynomial<C> h)

Annihilator for element modulo this ideal.

Parameters:

h - solvable polynomial

Returns:

annihilator of h with respect to this

isAnnihilator

public boolean isAnnihilator(GenSolvablePolynomial<C> h,
                             SolvableIdeal<C> A)

Test for annihilator of element modulo this ideal.

Parameters:

h - solvable polynomial

A - solvable ideal

Returns:

true, if A is the annihilator of h with respect to this

annihilator

public SolvableIdeal<C> annihilator(SolvableIdeal<C> H)

Annihilator for ideal modulo this ideal.

Parameters:

H - solvable ideal

Returns:

annihilator of H with respect to this

isAnnihilator

public boolean isAnnihilator(SolvableIdeal<C> H,
                             SolvableIdeal<C> A)

Test for annihilator of ideal modulo this ideal.

Parameters:

H - solvable ideal

A - solvable ideal

Returns:

true, if A is the annihilator of H with respect to this

inverse

public GenSolvablePolynomial<C> inverse(GenSolvablePolynomial<C> h)

Inverse for element modulo this ideal.

Parameters:

h - solvable polynomial

Returns:

inverse of h with respect to this, if defined

isUnit

public boolean isUnit(GenSolvablePolynomial<C> h)

Test if element is a unit modulo this ideal.

Parameters:

h - solvable polynomial

Returns:

true if h is a unit with respect to this, else false

commonZeroTest

public int commonZeroTest()

Ideal common zero test.

Returns:

-1, 0 or 1 if dimension(this) &eq; -1, 0 or ≥ 1.

isMaximal

public boolean isMaximal()

Test if this ideal is maximal.

Returns:

true, if this is maximal and not one, else false.

univariateDegrees

public java.util.List<java.lang.Long> univariateDegrees()

Univariate head term degrees.

Returns:

a list of the degrees of univariate head terms.

dimension

public Dimension dimension()

Ideal dimension.

Returns:

a dimension container (dim,maxIndep,list(maxIndep),vars).

constructUnivariate

public java.util.List<GenSolvablePolynomial<C>> constructUnivariate()

Construct univariate polynomials of minimal degree in all variables in zero dimensional ideal(G).

Returns:

list of univariate solvable polynomial of minimal degree in each variable in ideal(G)

constructUnivariate

public GenSolvablePolynomial<C> constructUnivariate(int i)

Construct univariate polynomial of minimal degree in variable i in zero dimensional ideal(G).

Parameters:

i - variable index.

Returns:

univariate solvable polynomial of minimal degree in variable i in ideal(G)