3. Lifting fiber

This package provides functions for manipulating lifting libers. Lifting fibers encode equidimensional algebraic varieties.
Before all, a BlackboxPolynomialAlgebra domain must have been built this way:

E,x[1],...,x[n]:= BlackboxPolynomialAlgebra(F,n);
where F is either the field of the rational numbers or a prime field. Note that it may work with other fields as soon as the characteristic is big enough and a factorization function exists in F[T]. A fiber encoding an r-equidimensional algebraic variety V is the following record:

ResolutionField, a field K over which the resolution is defined. There must exist a coercion from F to K.
LiftingSystem, a lifting system F for V, it is a sequence of blackbox polynomials.
PrimitiveElement, a linear form u in the x[i]'s over K, it is a multivariate polynomial in variables.
MinimalPolynomial, a sequence q of monic univariate polynomials in K[T], of size s.
Denominator, a sequence p of univariate polynomials in K[T] of size s.
Parametrization, a sequence w of size s of sequences of elements of K[T] of size n-r.
MagicPoint, a sequence P of elements in K of size r.
ChangeOfVariables, a record <LinearPart,AffinePart>, LinearPart is an invertible square matrix M of size and AffinePart is a vector b of size n. Booth have entries in K.
IsMultiple, boolean flag telling if V is multiple as a solution of F=0.
GenericTrace, generic trace of the deflation process.
ParentBbp points to E.

Let y be the new variables defined by x=M.y+b, the following properties hold:

V is a subvariety of the set of roots of F=0.
The variables y are in projective Noether position with respect to V.
y[1]=P[1],...,y[r]=P[r] defines a finite fiber V' of V.
The primitive element u separates the points of this fiber V'.
The elements of q are monic and squarefree.
The fiber V' is the union of the set of points described by the parametrizations:
q[l](T)=0, p[l](T)y[r+1]=v[l](T), ..., p[l](T)y[n]=v[n](T), for l in [1,...,s].

We say the fiber is isolated if V is an isolated subvariety in the set of roots of the system F=0. We say that the magic point is a lifting point if it satisfies the smoothess hypothesis of the fast deflation algorithm. The parametrization of is in the Kronecker presentation if p[l]=q'[l] for all l and is in the Shape-lemma if p[l]=1 for all l.

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3.1 Creation of lifting fibers

intrinsic: LiftingFiber () -> Cat: It returns the record format used to store lifting fibers.

intrinsic: WholeSpaceLF (E, K: GenericLinearChangeOfVariables:=true) -> .

E, a domain created by BlackboxPolynomialAlgebra.
K, a field.

It returns a lifting fiber of the ambient space with resolution field K.
Parameter: If GenericLinearChangeOfVariables is set then a generic affine change of the coordinates is performed.

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3.2 Getting properties

lf denotes a lifting fiber and llf a sequence of lifting fibers.

intrinsic: CodimensionLF (lf::Rec) -> RngIntElt: It returns the codimension of the variety encoded by lf.

intrinsic: CodimensionLF (llf::[]) -> RngIntElt: It returns the minimum of the codimensions of the elements of llf.
Error condition: llf must not be empty.

intrinsic: DegreeLF (lf::Rec) -> RngIntElt: It returns the degree of the variety represented by lf.

intrinsic: DegreeLF (llf::[]) -> RngIntElt: It returns the sum of the degrees of the element of llf.

intrinsic: DimensionLF (lf) -> RngIntElt: See RankLF.

intrinsic: HasKroneckerParametrizationLF (lf::Rec) -> BoolElt: It returns true if lf has a Kronecker parametrization.

intrinsic: HasKroneckerParametrizationLF (llf::[]) -> BoolElt: It returns true if all the elements of llf have a Kronecker parametrization.

intrinsic: HasShapeLemmaParametrizationLF (lf::Rec) -> BoolElt: It returns true if lf has a shape lemma parametrization.

intrinsic: HasShapeLemmaParametrizationLF (llf::[]) -> BoolElt: It returns true if all the elements of lf have a shape lemma parametrization.

intrinsic: IsEmptyLF (lf::Rec) -> BoolElt: It returns a boolean telling whether the variety represented by lf is empty or not.

intrinsic: IsEmptyLF (llf::[]) -> BoolElt: It returns a boolean telling whether all the elements of llf represent the empty variety or not.

intrinsic: IsMultipleLF (lf::Rec) -> BoolElt: It returns a boolean telling whether lf is a multiple component or not.

intrinsic: IsWholeSpaceLF (lf::Rec) -> BoolElt: It returns whether lf represents the ambient space or not.

intrinsic: IsWholeSpaceLF (llf::[]) -> BoolElt: It returns whether at least one of the elements of llf represents the ambient space or not.

intrinsic: NumberOfVariablesLF (lf::Rec) -> RngIntElt: It returns the dimension of the ambient space in which lf lives.

intrinsic: NumberOfFactorsLF (lf::Rec) -> RngIntElt: It returns the number of factors lf, that is the cardinal of lf`MinimalPolynomial.

intrinsic: ParentBbpLF (lf::Rec) -> Cat: It returns the Multivariate Polynomial Algebra associated to the blackbox polynomial domain of the lifting system of lf.

intrinsic: RankLF (lf::Rec) -> RngIntElt: It returns the dimension of the variety encoded by lf.

intrinsic: RankLF (llf::[]) -> RngIntElt: It returns the maximum of the dimensions of the elements of llf.
Error condition: llf must not be empty.

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3.3 Basic operations

lf denotes a lifting fiber and llf a sequence of lifting fibers.

intrinsic: ChangeResolutionFieldLF (~lf::Rec, K)

lf, a LiftingFiber.
K, a field.

If there exists a coercion from lf`ResolutionField to K, then it modifies lf to be a resolution over K.
Error condition: There must exist a coercion from lf`ResolutionField to K.

intrinsic: ChangeResolutionFieldLF (~llf::[], K): It iterates ChangeResolutionFieldLF over llf.

intrinsic: ChangeResolutionFieldLF (~lf::Rec, K, f)

lf, a LiftingFiber.
K, a field.
f, a homomorphism from lf`ResolutionField to K.

Applies f on lf`ResolutionField to coerce it to K via f, so that in return lf is a resolution over K.

intrinsic: ChangeResolutionFieldLF (~llf::[], K, f): It iterates ChangeResolutionFieldLF over llf.

intrinsic: MakeKroneckerParametrizationLF (~lf::Rec): lf is changed to a Kronecker parametrization.

intrinsic: MakeKroneckerParametrizationLF (~llf::[]): It iterates MakeKroneckerParametrizationLF over each element of llf.

intrinsic: MakeMonicLF (~lf::Rec): It makes lf`MinimalPolynomial monic.

intrinsic: MakeMonicLF (~llf::[]): It iterates MakeMonicLF over each element of llf.

intrinsic: MakeShapeLemmaParametrizationLF (~lf::Rec): It changes the parametrization of lf into a Shape Lemma form.
Error condition: If lf`Denominator is not invertible modulo lf`MinimalPolynomial. then lf contains a string in return.

intrinsic: MakeShapeLemmaParametrizationLF (~llf::[]): It iterates MakeShapeLemmaParametrizationLF over each element of llf.

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3.4 Changes of fibers

intrinsic: ChangeAlgebraicVariablesLF (~lf::Rec, N, c)

lf, a LiftingFiber of dimension r in a n-dimensional space.
N, a square matrix of size n-r with rational numbers entries.
c, a n-r-vector with rational numbers entries.

Let M and b be lf`ChangeOfVariables, such that x=My+b. The procedure changes the coordinates of lf in the following way: M:=M.(Idr|N) and b:=M.(0|c) + b. This change of algebraic variables is applied in consequence to the parametrization and the primitive element so that lf remains consistent.
Error condition: The procedure raises an error if N is not invertible.

intrinsic: ChangeAlgebraicVariablesLF (~llf::[], N, c): It applies ChangeAlgebraicVariablesLF to each element of llf.

intrinsic: ChangeBackAlgebraicVariablesLF (~lf::Rec)

lf, a LiftingFiber.

In return lf has the identity for the part of the change of variables corresponding to the algebraic variables.

intrinsic: ChangeBackAlgebraicVariablesLF (~llf::[]): It applies ChangeBackAlgebraicVariablesLF to each element of llf.

intrinsic: ChangeBackFreeVariablesLF (~lf::Rec)

lf, a LiftingFiber.

In return lf has the identity in the part of its change of variables corresponding to the free variables.

intrinsic: ChangeBackFreeVariablesLF (~llf::[]): It iterates ChangeBackFreeVariablesLF on each element of llf.

intrinsic: ChangePrimitiveElementLF (~lf::Rec, u)

lf, a LiftingFiber of dimension r in a n-dimensional space.
u, a linear form given as a multivariate polynomial.

It Changes lf`PrimitiveElement to u, if it is actually a primitive element.
Error condition: The procedure returns "Bad primitive element" in lf if u is not a primitive element.

intrinsic: ChangePrimitiveElementLF (~llf::[], u): It applies ChangePrimitiveElementLF to each element of llf.

intrinsic: TranslateKthFreeVariableToZeroLF (~lf::Rec, k)

lf, a LiftingFiber.

Let y[k] be the kth free variable of lf, the procedure modifies lf`ChangeOfVariables replacing y[k] by y[k]+lf`MagicPoint[k]. Then lf`MagicPoint[k] is set to 0.

intrinsic: TranslateKthFreeVariableToZeroLF (~llf::[], k): It iterates TranslateKthFreeVariableToZeroLF on each element of llf.

intrinsic: TranslateAllFreeVariablesToZeroLF (~lf::Rec): It applies TranslateKthFreeVariableToZeroLF to all the free variables of lf.

intrinsic: TranslateAllFreeVariablesToZeroLF (~llf::[]): It applies TranslateAllFreeVariablesToZeroLF to each element of llf.

intrinsic: TranslateLastFreeVariableToZeroLF (~lf::Rec): It applies TranslateKthFreeVariableToZeroLF to all the last free variables of lf.

intrinsic: TranslateLastFreeVariableToZeroLF (~llf::[]): It applies TranslateLastFreeVariableToZeroLF to each element of llf.

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3.5 Evaluation

lf denotes a lifting fiber and llf a sequence of lifting fibers.

intrinsic: EvaluateLF (lf::Rec, f::Rec, x) -> .

intrinsic: EvaluateLF (lf::Rec, f::[Rec], x::[]) -> .

lf, a LiftingFiber.
f, a sequence of black box polynomials.
x, a sequence of values.

It returns the sequence of the values of f evaluated on the point x.
Error conditions:

#x must be equal to NumberOfVariablesLF(lf).
BaseRing(ParentBbpLF(lf)) must be coercible to Universe(x).

intrinsic: VerifyLF (lf::Rec: Strategy:="Probabilistic") -> BoolElt

It returns true iff the parametrization of lf satisfies its lifting system modulo its MinimalPolynomial.
Parameter: Strategy, string:

"", the verification is perfomed over lf`ResolutionField.
"Probabilistic", the verification is probabilistic. Computations are done modulo a random prime number if the resolution field is the field of the rational numbers. If the resolution field is a rational function field, the variables are specialized at random.

intrinsic: VerifyLF (llf::[]: Strategy:="Probabilistic") -> BoolElt: It tells whether all the elements of llf satisfies their lifting systems.

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3.6 Splittings

intrinsic: SplitLF (lf::Rec, f::[Rec]) -> Rec,Rec

intrinsic: SplitLF (lf::Rec, f::Rec) -> Rec,Rec

lf, a lifting fiber.
f, an element of BlackboxPolynomialAlgebra.

It returns two fibers lfz, lfnz. The first one represents the points of lf satisfying f=0 and the second one the other points.

intrinsic: CleanLF (~lf::Rec, ineqs::Rec): Removes the points of lf satisfying ineqs=0.

intrinsic: CleanLF (~lf::Rec, ineqs::[]): Removes the points of lf satisfying ineqs=0.

intrinsic: CleanLF (~llf::[], ineqs): It iterates CleanLF over each element of llf.

intrinsic: FactorizationLF (~lf::Rec): lf is factorized.
Error condition: The parametrization must be shape lemma.

intrinsic: CombineLF (~lf::Rec: Parametrization:="Unknown"): It combines the factors of lf. The parameter can be "Unknown" (default), "ShapeLemma" or "Kronecker" according to the properties known about lf.

intrinsic: MergeLF (llf::[]) -> Rec

It merges the elements of llf into one.
Error conditions:

The elements of llf must share the same ResolutionField, LiftingSystem, PrimitiveElement, MagicPoint, ChangeOfVariables, GenericTrace and ParentBbp.
llf must not be empty.

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3.7 Printing

intrinsic: PrintLF (lf::Rec): It prints lf on stdout.

intrinsic: PrintLF (llf::[]): Prints llf on stdout.

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