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#include <tropicalStrategy.h>
Public Member Functions | |
tropicalStrategy (const ideal I, const ring r, const bool completelyHomogeneous=true, const bool completeSpace=true) | |
Constructor for the trivial valuation case. More... | |
tropicalStrategy (const ideal J, const number p, const ring s) | |
Constructor for the non-trivial valuation case p is the uniformizing parameter of the valuation. More... | |
tropicalStrategy (const tropicalStrategy ¤tStrategy) | |
copy constructor More... | |
~tropicalStrategy () | |
destructor More... | |
tropicalStrategy & | operator= (const tropicalStrategy ¤tStrategy) |
assignment operator More... | |
bool | isValuationTrivial () const |
bool | isValuationNonTrivial () const |
ring | getOriginalRing () const |
returns the polynomial ring over the field with valuation More... | |
ideal | getOriginalIdeal () const |
returns the input ideal over the field with valuation More... | |
ring | getStartingRing () const |
returns the polynomial ring over the valuation ring More... | |
ideal | getStartingIdeal () const |
returns the input ideal More... | |
int | getExpectedAmbientDimension () const |
int | getExpectedDimension () const |
returns the expected Dimension of the polyhedral output More... | |
number | getUniformizingParameter () const |
returns the uniformizing parameter in the valuation ring More... | |
ring | getShortcutRing () const |
gfan::ZCone | getHomogeneitySpace () const |
returns the homogeneity space of the preimage ideal More... | |
bool | homogeneitySpaceContains (const gfan::ZVector &v) const |
returns true, if v is contained in the homogeneity space; false otherwise More... | |
bool | restrictToLowerHalfSpace () const |
returns true, if valuation non-trivial, false otherwise More... | |
gfan::ZVector | adjustWeightForHomogeneity (gfan::ZVector w) const |
Given weight w, returns a strictly positive weight u such that an ideal satisfying the valuation-sepcific homogeneity conditions is weighted homogeneous with respect to w if and only if it is homogeneous with respect to u. More... | |
gfan::ZVector | adjustWeightUnderHomogeneity (gfan::ZVector v, gfan::ZVector w) const |
Given strictly positive weight w and weight v, returns a strictly positive weight u such that on an ideal that is weighted homogeneous with respect to w the weights u and v coincide. More... | |
gfan::ZVector | negateWeight (const gfan::ZVector &w) const |
ring | getShortcutRingPrependingWeight (const ring r, const gfan::ZVector &w) const |
If valuation trivial, returns a copy of r with a positive weight prepended, such that any ideal homogeneous with respect to w is homogeneous with respect to that weight. More... | |
bool | reduce (ideal I, const ring r) const |
reduces the generators of an ideal I so that the inequalities and equations of the Groebner cone can be read off. More... | |
void | pReduce (ideal I, const ring r) const |
std::pair< poly, int > | checkInitialIdealForMonomial (const ideal I, const ring r, const gfan::ZVector &w=0) const |
If given w, assuming w is in the Groebner cone of the ordering on r and I is a standard basis with respect to that ordering, checks whether the initial ideal of I with respect to w contains a monomial. More... | |
ideal | computeStdOfInitialIdeal (const ideal inI, const ring r) const |
given generators of the initial ideal, computes its standard basis More... | |
ideal | computeWitness (const ideal inJ, const ideal inI, const ideal I, const ring r) const |
suppose w a weight in maximal groebner cone of > suppose I (initially) reduced standard basis w.r.t. More... | |
ideal | computeLift (const ideal inJs, const ring s, const ideal inIr, const ideal Ir, const ring r) const |
std::pair< ideal, ring > | computeFlip (const ideal Ir, const ring r, const gfan::ZVector &interiorPoint, const gfan::ZVector &facetNormal) const |
given an interior point of a groebner cone computes the groebner cone adjacent to it More... | |
Private Member Functions | |
ring | copyAndChangeCoefficientRing (const ring r) const |
ring | copyAndChangeOrderingWP (const ring r, const gfan::ZVector &w, const gfan::ZVector &v) const |
ring | copyAndChangeOrderingLS (const ring r, const gfan::ZVector &w, const gfan::ZVector &v) const |
bool | checkForUniformizingBinomial (const ideal I, const ring r) const |
if valuation non-trivial, checks whether the generating system contains p-t otherwise returns true More... | |
bool | checkForUniformizingParameter (const ideal inI, const ring r) const |
if valuation non-trivial, checks whether the genearting system contains p otherwise returns true More... | |
int | findPositionOfUniformizingBinomial (const ideal I, const ring r) const |
void | putUniformizingBinomialInFront (ideal I, const ring r, const number q) const |
Private Attributes | |
ring | originalRing |
polynomial ring over a field with valuation More... | |
ideal | originalIdeal |
input ideal, assumed to be a homogeneous prime ideal More... | |
int | expectedDimension |
the expected Dimension of the polyhedral output, i.e. More... | |
gfan::ZCone | linealitySpace |
the homogeneity space of the Grobner fan More... | |
ring | startingRing |
polynomial ring over the valuation ring extended by one extra variable t More... | |
ideal | startingIdeal |
preimage of the input ideal under the map that sends t to the uniformizing parameter More... | |
number | uniformizingParameter |
uniformizing parameter in the valuation ring More... | |
ring | shortcutRing |
polynomial ring over the residue field More... | |
bool | onlyLowerHalfSpace |
true if valuation non-trivial, false otherwise More... | |
gfan::ZVector(* | weightAdjustingAlgorithm1 )(const gfan::ZVector &w) |
A function such that: Given weight w, returns a strictly positive weight u such that an ideal satisfying the valuation-sepcific homogeneity conditions is weighted homogeneous with respect to w if and only if it is homogeneous with respect to u. More... | |
gfan::ZVector(* | weightAdjustingAlgorithm2 )(const gfan::ZVector &v, const gfan::ZVector &w) |
A function such that: Given strictly positive weight w and weight v, returns a strictly positive weight u such that on an ideal that is weighted homogeneous with respect to w the weights u and v coincide. More... | |
bool(* | extraReductionAlgorithm )(ideal I, ring r, number p) |
A function that reduces the generators of an ideal I so that the inequalities and equations of the Groebner cone can be read off. More... | |
Definition at line 36 of file tropicalStrategy.h.
tropicalStrategy::tropicalStrategy | ( | const ideal | I, |
const ring | r, | ||
const bool | completelyHomogeneous = true , |
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const bool | completeSpace = true |
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Constructor for the trivial valuation case.
Initializes all relevant structures and information for the trivial valuation case, i.e.
computing a tropical variety without any valuation.
Definition at line 138 of file tropicalStrategy.cc.
Constructor for the non-trivial valuation case p is the uniformizing parameter of the valuation.
Definition at line 277 of file tropicalStrategy.cc.
tropicalStrategy::tropicalStrategy | ( | const tropicalStrategy & | currentStrategy | ) |
copy constructor
Definition at line 316 of file tropicalStrategy.cc.
tropicalStrategy::~tropicalStrategy | ( | ) |
destructor
Definition at line 346 of file tropicalStrategy.cc.
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Given weight w, returns a strictly positive weight u such that an ideal satisfying the valuation-sepcific homogeneity conditions is weighted homogeneous with respect to w if and only if it is homogeneous with respect to u.
Definition at line 248 of file tropicalStrategy.h.
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Given strictly positive weight w and weight v, returns a strictly positive weight u such that on an ideal that is weighted homogeneous with respect to w the weights u and v coincide.
Definition at line 258 of file tropicalStrategy.h.
if valuation non-trivial, checks whether the generating system contains p-t otherwise returns true
Definition at line 814 of file tropicalStrategy.cc.
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if valuation non-trivial, checks whether the genearting system contains p otherwise returns true
Definition at line 867 of file tropicalStrategy.cc.
std::pair< poly, int > tropicalStrategy::checkInitialIdealForMonomial | ( | const ideal | I, |
const ring | r, | ||
const gfan::ZVector & | w = 0 |
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) | const |
If given w, assuming w is in the Groebner cone of the ordering on r and I is a standard basis with respect to that ordering, checks whether the initial ideal of I with respect to w contains a monomial.
If no w is given, assuming that I is already an initial form of some ideal, checks whether I contains a monomial. In both cases returns a monomial, if it contains one, returns NULL otherwise.
Definition at line 496 of file tropicalStrategy.cc.
std::pair< ideal, ring > tropicalStrategy::computeFlip | ( | const ideal | Ir, |
const ring | r, | ||
const gfan::ZVector & | interiorPoint, | ||
const gfan::ZVector & | facetNormal | ||
) | const |
given an interior point of a groebner cone computes the groebner cone adjacent to it
Definition at line 762 of file tropicalStrategy.cc.
ideal tropicalStrategy::computeLift | ( | const ideal | inJs, |
const ring | s, | ||
const ideal | inIr, | ||
const ideal | Ir, | ||
const ring | r | ||
) | const |
Definition at line 688 of file tropicalStrategy.cc.
given generators of the initial ideal, computes its standard basis
Definition at line 656 of file tropicalStrategy.cc.
ideal tropicalStrategy::computeWitness | ( | const ideal | inJ, |
const ideal | inI, | ||
const ideal | I, | ||
const ring | r | ||
) | const |
suppose w a weight in maximal groebner cone of > suppose I (initially) reduced standard basis w.r.t.
> and inI initial forms of its elements w.r.t. w suppose inJ elements of initial ideal that are homogeneous w.r.t w returns J elements of ideal whose initial form w.r.t. w are inI in particular, if w lies also inthe maximal groebner cone of another ordering >' and inJ is a standard basis of the initial ideal w.r.t. >' then the returned J will be a standard baiss of the ideal w.r.t. >'
change ground ring into finite field and map the data into it
Compute a division with remainder over the finite field and map the result back to r
Compute the normal forms
Definition at line 574 of file tropicalStrategy.cc.
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Definition at line 564 of file tropicalStrategy.cc.
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Definition at line 734 of file tropicalStrategy.cc.
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Definition at line 704 of file tropicalStrategy.cc.
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Definition at line 191 of file tropicalStrategy.h.
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returns the expected Dimension of the polyhedral output
Definition at line 199 of file tropicalStrategy.h.
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returns the homogeneity space of the preimage ideal
Definition at line 222 of file tropicalStrategy.h.
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returns the input ideal over the field with valuation
Definition at line 167 of file tropicalStrategy.h.
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returns the polynomial ring over the field with valuation
Definition at line 158 of file tropicalStrategy.h.
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Definition at line 213 of file tropicalStrategy.h.
ring tropicalStrategy::getShortcutRingPrependingWeight | ( | const ring | r, |
const gfan::ZVector & | w | ||
) | const |
If valuation trivial, returns a copy of r with a positive weight prepended, such that any ideal homogeneous with respect to w is homogeneous with respect to that weight.
If valuation non-trivial, changes the coefficient ring to the residue field.
Definition at line 448 of file tropicalStrategy.cc.
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returns the input ideal
Definition at line 185 of file tropicalStrategy.h.
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returns the polynomial ring over the valuation ring
Definition at line 176 of file tropicalStrategy.h.
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returns the uniformizing parameter in the valuation ring
Definition at line 207 of file tropicalStrategy.h.
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returns true, if v is contained in the homogeneity space; false otherwise
Definition at line 230 of file tropicalStrategy.h.
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Definition at line 149 of file tropicalStrategy.h.
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Definition at line 144 of file tropicalStrategy.h.
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tropicalStrategy & tropicalStrategy::operator= | ( | const tropicalStrategy & | currentStrategy | ) |
assignment operator
Definition at line 363 of file tropicalStrategy.cc.
void tropicalStrategy::pReduce | ( | ideal | I, |
const ring | r | ||
) | const |
Definition at line 432 of file tropicalStrategy.cc.
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bool tropicalStrategy::reduce | ( | ideal | I, |
const ring | r | ||
) | const |
reduces the generators of an ideal I so that the inequalities and equations of the Groebner cone can be read off.
Definition at line 416 of file tropicalStrategy.cc.
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returns true, if valuation non-trivial, false otherwise
Definition at line 238 of file tropicalStrategy.h.
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the expected Dimension of the polyhedral output, i.e.
the dimension of the ideal if valuation trivial or the dimension of the ideal plus one if valuation non-trivial (as the output is supposed to be intersected with a hyperplane)
Definition at line 53 of file tropicalStrategy.h.
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A function that reduces the generators of an ideal I so that the inequalities and equations of the Groebner cone can be read off.
Definition at line 98 of file tropicalStrategy.h.
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the homogeneity space of the Grobner fan
Definition at line 57 of file tropicalStrategy.h.
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true if valuation non-trivial, false otherwise
Definition at line 78 of file tropicalStrategy.h.
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input ideal, assumed to be a homogeneous prime ideal
Definition at line 46 of file tropicalStrategy.h.
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polynomial ring over a field with valuation
Definition at line 42 of file tropicalStrategy.h.
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polynomial ring over the residue field
Definition at line 73 of file tropicalStrategy.h.
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preimage of the input ideal under the map that sends t to the uniformizing parameter
Definition at line 65 of file tropicalStrategy.h.
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polynomial ring over the valuation ring extended by one extra variable t
Definition at line 61 of file tropicalStrategy.h.
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uniformizing parameter in the valuation ring
Definition at line 69 of file tropicalStrategy.h.
A function such that: Given weight w, returns a strictly positive weight u such that an ideal satisfying the valuation-sepcific homogeneity conditions is weighted homogeneous with respect to w if and only if it is homogeneous with respect to u.
Definition at line 86 of file tropicalStrategy.h.
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A function such that: Given strictly positive weight w and weight v, returns a strictly positive weight u such that on an ideal that is weighted homogeneous with respect to w the weights u and v coincide.
Definition at line 93 of file tropicalStrategy.h.