5. Equidimensional decomposition

This section provides functionalities to handle equidimensional decomposition. The verbose flag is GeometricSolveVerbose.

5.1 Inclusion of components

intrinsic: IsIncludedIrreducibleLF (lf1, l, lf2::Rec) -> BoolElt

• lf1 is a fiber.
• l is an integer.
• lf2 is an isolated lifting fiber.

The function tells whether the lth irreducible factor of lf1 is included in lf2 or not. If it is detected that lf2 is not an isolated lifting fiber then a string is returned.

intrinsic: IsIncludedIrreducibleLF (lf, l, llf::SeqEnum) -> BoolElt

The function iterates IsIncludedIrreducibleLF over each element of llf and tells whether lf[l] is included in llf.

5.2 Set difference

intrinsic: DifferenceLF (~lf1, lf2::Rec)

• lf1 is a fiber.
• lf2 is an isolated lifting fiber.

In return lf1 contains its only components that are not included in lf2.
Error condition: If it is detected that lf2 is not an isolated lifting fiber then lf1 contains a string in return.

intrinsic: DifferenceLF (~lf, llf::SeqEnum)

The procedure iterates DifferenceLF over each element of llf, so that lf only contains its components that are not included in llf in return.

5.3 Minimization

intrinsic: MinimizeLLF (~llf: RemoveMultipleComponents:=false)

llf is an equidimenstional decomposition that may be redundant. In return llf does not contain redundant components. It is important that llf[i] contains the components of dimension i+1. In case of bugs llf may contain a string in return.

5.4 Intersection

intrinsic: IntersectLF (llf::SeqEnum, f::Rec, h::[Rec]: RemoveMultipleComponents:=false) -> SeqEnum

• llf, a minimal sequence of sequences lifting fibers encoding an equidimensional decomposition of an algebraic variety.
• f, a blackbox polynomial.
• h, a sequence of blackbox polynomials.

It returns the minimal equidimensional decomposition for the intersection of the input variety given by llf with the hypersurface defined by f=0 and outside h=0.
Error condition: In case of problems (bug or unlucky choices) a string is returned.