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ParametricGroebnerBases.CGB — Method
CGB(F::Vector{RE}, reduced=true) where {RE<:MPoly}

Computes a comprehensive parametric Gröbner basis of the ideal generated by F. In other words specialize.(G, Ref(α)) is a Gröbner basis for any vector α of appropriate length.

If the option reduced=true, then specialize.(G, Ref(α)) always contains the reduced Gröbner basis of the ideal generated by specialize.(F, Ref(α)).

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ParametricGroebnerBases.CGS — Method
CGS(F::Vector{RE}, reduced=true) where {RE <: MPoly}

Function to compute reduced Gröbner systems. Returns a vector of triples of the form (S, [h], G). For any vector α such that specialize.(S, Ref(α)) == [0, 0, ..., 0] and specialize(h, α) != 0, specialize.(G, Ref(α)) is a reduced Gröbner basis.

Takes the argument reduced. If set to false, the returned Gröbner basis will not be reduced.

See also [CGS_simple]

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ParametricGroebnerBases.CGS_faithful — Method
CGS_faithful(F::Vector{RE}, reduced=true) where {RE<:MPoly}

Function to compute faithful, reduced Gröbner systems. Returns a vector of triples of the form (S, [h], G). For any vector α such that specialize.(S, α) == [0, 0, ..., 0] and specialize(h, α) != 0, specialize.(G, Ref(α)) is a reduced Gröbner basis. Furthermore, each g ∈ G will belong to the ideal generated by F.

Takes the argument reduced. If set to false, the returned Gröbner basis will not be reduced.

See also CGS_simple

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ParametricGroebnerBases.CGS_simple — Method
CGS_simple(F::Vector{RE}) where {RE <: MPoly}

Simple function to compute Gröbner systems. Returns a vector of triples of the form (S, [h], G). For any vector α such that specialize.(S, Ref(α)) == [0, 0, ..., 0] and specialize(h, α) != 0, specialize.(G, Ref(α)) is a Gröbner basis.

See also CGS

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ParametricGroebnerBases.pseudo_reduce — Method
pseudo_reduce(f::RE, G, reduced = false) where {RE<:MPoly}

Pseudo-reduces $f$ modulo $G = \{g_1, g_2, \dots, g_n\}$. Pseudo-reduction writes

\[c f = r + \sum_{i=1}^n f_i g_i\]

where $c$ is a product of leading coefficients of $G$, $\operatorname{lm}(f_i g_i) \leq \operatorname{lm}(f)$, and no term of $r$ is divisible by any leading monomial of $G$.

The result of pseudo_reduce(f, G) is the pair (c, r).

Takes the argument reduced. If set to true, the function will remove factors in r coming from leading coefficients of $G$.

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